[ ]
[ ] Linear Programming
Frequently Asked Questions
Optimization Technology Center of
Northwestern University and Argonne National Laboratory
[ ] Posted monthly to Usenet newsgroup sci.op-research
World Wide Web version:
http://www.mcs.anl.gov/home/otc/Guide/faq/linear-programming-faq.html
Plain-text version:
ftp://rtfm.mit.edu/pub/usenet/sci.answers/linear-programming-faq
Date of this version: August 1, 1997
* Q1. "What is Linear Programming?"
* Q2. "Where is there good software to solve LP problems?"
* Q3. "Oh, and we also want to solve it as an integer program."
* Q4. "I wrote an optimization code. Where are some test models?"
* Q5. "What is MPS format?"
* Q6. Topics briefly covered:
o Q6.1: "What is a modeling language?"
o Q6.2: "How do I diagnose an infeasible LP model?"
o Q6.3: "I want to know the specific constraints that contradict
each other."
o Q6.4: "I just want to know whether or not a feasible solution
*exists*."
o Q6.5: "I have an LP, except it's got several objective functions."
o Q6.6: "I have an LP that has large almost-independent matrix
blocks that are linked by a few constraints. Can I take advantage
of this?"
o Q6.7: "I am looking for an algorithm to compute the convex hull of
a finite number of points in n-dimensional space."
o Q6.8: "Are there any parallel LP codes?"
o Q6.9: "What software is there for Network models?"
o Q6.10: "What software is there for the Traveling Salesman Problem
(TSP)?"
o Q6.11: "What software is there for the Knapsack Problem?"
o Q6.12: "What software is there for Stochastic Programming?"
o Q6.13: "I need to do post-optimal analysis."
o Q6.14: "Do LP codes require a starting vertex?"
o Q6.15: "How can I combat cycling in the Simplex algorithm?"
* Q7. "What references and Web links are there in this field?"
* Q8. "How do I access the Netlib server?"
* Q9. "Who maintains this FAQ list?"
See also the following pages
pertaining to mathematical programming and optimization modeling:
* The related Nonlinear Programming FAQ.
* The NEOS Guide to optimization models and software.
* The Decision Tree for Optimization Software by H.D. Mittelmann and P.
Spellucci.
* Jiefeng Xu's List of Interesting Optimization Codes in the Public
Domain.
* Software for Optimization: A Buyer's Guide by Robert Fourer.
* Harvey Greenberg's Mathematical Programming Glossary.
[ ]
Q1. "What is Linear Programming?"
A: (For rigorous definitions and theory, which are beyond the scope of this
document, the interested reader is referred to the many LP textbooks in
print, a few of which are listed in the references section.)
A Linear Program (LP) is a problem that can be expressed as follows (the
so-called Standard Form):
minimize cx
subject to Ax = b
x >= 0
where x is the vector of variables to be solved for, A is a matrix of known
coefficients, and c and b are vectors of known coefficients. The expression
"cx" is called the objective function, and the equations "Ax=b" are called
the constraints. All these entities must have consistent dimensions, of
course, and you can add "transpose" symbols to taste. The matrix A is
generally not square, hence you don't solve an LP by just inverting A.
Usually A has more columns than rows, and Ax=b is therefore quite likely to
be under-determined, leaving great latitude in the choice of x with which to
minimize cx.
The word "Programming" is used here in the sense of "planning"; the
necessary relationship to computer programming was incidental to the choice
of name. Hence the phrase "LP program" to refer to a piece of software is
not a redundancy, although I tend to use the term "code" instead of
"program" to avoid the possible ambiguity.
Although all linear programs can be put into the Standard Form, in practice
it may not be necessary to do so. For example, although the Standard Form
requires all variables to be non-negative, most good LP software allows
general bounds l <= x <= u, where l and u are vectors of known lower and
upper bounds. Individual elements of these bounds vectors can even be
infinity and/or minus-infinity. This allows a variable to be without an
explicit upper or lower bound, although of course the constraints in the
A-matrix will need to put implied limits on the variable or else the problem
may have no finite solution. Similarly, good software allows b1 <= Ax <= b2
for arbitrary b1, b2; the user need not hide inequality constraints by the
inclusion of explicit "slack" variables, nor write Ax >= b1 and Ax <= b2 as
two separate constraints. Also, LP software can handle maximization problems
just as easily as minimization (in effect, the vector c is just multiplied
by -1).
The importance of linear programming derives in part from its many
applications (see further below) and in part from the existence of good
general-purpose techniques for finding optimal solutions. These techniques
take as input only an LP in the above Standard Form, and determine a
solution without reference to any information concerning the LP's origins or
special structure. They are fast and reliable over a substantial range of
problem sizes and applications.
Two families of solution techniques are in wide use today. Both visit a
progressively improving series of trial solutions, until a solution is
reached that satisfies the conditions for an optimum. Simplex methods,
introduced by Dantzig about 50 years ago, visit "basic" solutions computed
by fixing enough of the variables at their bounds to reduce the constraints
Ax = b to a square system, which can be solved for unique values of the
remaining variables. Basic solutions represent extreme boundary points of
the feasible region defined by Ax = b, x >= 0, and the simplex method can be
viewed as moving from one such point to another along the edges of the
boundary. Barrier or interior-point methods, by contrast, visit points
within the interior of the feasible region. These methods derive from
techniques for nonlinear programming that were developed and popularized in
the 1960s by Fiacco and McCormick, but their application to linear
programming dates back only to Karmarkar's innovative analysis in 1984.
The related problem of integer programming (or integer linear programming,
strictly speaking) requires some or all of the variables to take integer
(whole number) values. Integer programs (IPs) often have the advantage of
being more realistic than LPs, but the disadvantage of being much harder to
solve. The most widely used general-purpose techniques for solving IPs use
the solutions to a series of LPs to manage the search for integer solutions
and to prove optimality. Thus most IP software is built upon LP software,
and this FAQ applies to problems of both kinds.
Linear and integer programming have proved valuable for modeling many and
diverse types of problems in planning, routing, scheduling, assignment, and
design. Industries that make use of LP and its extensions include
transportation, energy, telecommunications, and manufacturing of many kinds.
A sampling of applications can be found in many LP textbooks, in books on LP
software systems, and among the application cases in the journal Interfaces.
[ ]
Q2. "Where is there good software to solve LP problems?"
A: Thanks to the advances in computing of the past decade, linear programs
in a few thousand variables and constraints are nowadays viewed as "small".
Problems having tens or hundreds of thousands of continuous variables are
regularly solved; tractable integer programs are necessarily smaller, but
are still commonly in the hundreds or thousands of variables and
constraints. The computers of choice for linear and integer programming
applications are Pentium-based PCs and the several varieties of Unix
workstations.
There is more to linear programming than optimal solutions and
number-crunching, however. This can be appreciated by observing that modern
LP software comes in two related but very different kinds of packages:
* Algorithmic codes are devoted to finding optimal solutions to specific
linear programs. A code takes as input a compact listing of the LP
constraint coefficients (the A, b, c and related values in the standard
form) and produces as output a similarly compact listing of optimal
solution values and related information.
* Modeling systems are designed to help people formulate LPs and analyze
their solutions. An LP modeling system takes as input a description of
a linear program in a form that people find reasonably natural and
convenient, and allows the solution output to be viewed in similar
terms; conversion to the forms requried by algorithmic codes is done
automatically. The collection of statement forms for the input is often
called a modeling language.
Most modeling systems support a variety of algorithmic codes, while the more
popular codes can be used with many different modeling systems. Because
packages of the two kinds are often bundled for convenience of marketing or
operation, the distinction between them is sometimes obscured, but it is
important to keep in mind when attempting to sort through the many
alternatives available.
Large-scale LP algorithmic codes rely on general-structure sparse matrix
techniques and numerous other refinements developed through years of
experience. The fastest and most reliable codes thus represent considerable
development effort, and tend to be expensive except in very limited
demonstration or "student" versions. Those codes that are free -- to all, or
at least for research and teaching -- tend to be somewhat less robust,
though they are still useful for many problems. The ability of a code to
solve any particular class of problems cannot easily be predicted from
problem size alone; some experimentation is usually necessary to establish
difficulty.
Large-scale LP modeling systems are commercial products virtually without
exception, and tend to be as expensive as the commercial algorithmic codes
(again with the exception of small demo versions). They vary so greatly in
design and capability that a description in words is adequate only to make a
preliminary decision among them; your ultimate choice is best guided by
using each candidate to formulate a model of interest.
Listed below are summary descriptions of available free codes, and a
tabulation of many commercial codes and modeling systems for linear (and
integer) programming. A list of free demos of commercial software appears at
the end of this section.
Another useful source of information is the Optimization Software Guide by
Jorge More' and Stephen Wright, available from SIAM Books. It contains
references to about 75 available software packages (not all of them just
LP), and goes into more detail than is possible in this FAQ; see in
particular the sections on "linear programming" and on "modeling languages
and optimization systems." An updated Web version of this book is available
on the NEOS Guide. Another good soruce of feature summaries and contact
information is the Linear Programming Software Survey compiled by OR/MS
Today (which also has the largest selection of advertisements for
optimization software). Much information can also be obtained through the
web sites of optimization software developers, many of which are identified
in the writeup and tables below.
To provide some idea of the relative performance of LP codes, a Web page of
pointers to benchmarks for optimization software is being compiled by Hans
Mittelmann of Arizona State University. It currently includes tests of
several public-domain simplex and interior-point implementations. When
evaluating any performance comparison, however, whether performed by a
customer, vendor, or disinterested third party, keep in mind that all
high-quality codes provide options that offer superior performance on
certain difficult kinds of LP or IP problems. Benchmark studies of the
"default settings" of codes will fail to reflect the power of the optional
settings that are available.
"Free" codes
Some of these programs require registration or payment for some (especially
commercial) uses. Conditions of use are also subject to change. It is a good
practice to check a code's "readme" file or introductory documentation for
restrictions before committing to use it.
Based on the simplex method:
There is an ftp-able code, written in C, called lp_solve that its author
(Michel Berkelaar, email at michel@es.ele.tue.nl) says has solved models
with up to 30,000 variables and 50,000 constraints. The author requests that
people retrieve it from ftp://ftp.es.ele.tue.nl/pub/lp_solve (numerical
address at last check: 131.155.20.126). There is an older version to be
found in the Usenet archives, but it contains bugs that have been fixed in
the meantime, and hence is unsupported. The author also made available a
program that converts data files from MPS-format into lp_solve's own input
format; it's in the same directory, in file mps2eq_0.2.tar.Z. The
documentation states that it is not public domain, and the author wants to
discuss it with would-be commercial users. As an editorial opinion, I must
state that difficult models will give lp_solve trouble; it's not as good as
a commercial code. But for someone who isn't sure what kind of LP code is
needed, it represents a reasonable first try.
LP-Optimizer is a simplex-based code for linear and integer programs,
written by Markus Weidenauer (nc-weidenma@netcologne.de). Free Borland
Pascal 7.0 source is available for downloading, as are executables for DOS
and OS/2.
SoPlex is an object-oriented implementation of the primal and dual simplex
algorithms, developed by Roland Wunderling. Source code is available free
for research uses at noncommercial and academic institutions.
Among the SLATEC library routines is a Fortran sparse implementation of the
simplex method, SPLP, at ftp://netlib2.cs.utk.edu/slatec/src/splp.f. Its
documentation states that it can solve LP models of "at most a few thousand
constraints and variables".
Based on interior-point methods:
The Optimization Technology Center at Argonne and Northwestern has developed
the interior-point code PCx. This code can be downloaded directly from the
PCx home page; it is freely available, except that you must contact Argonne
if you want to include it in a product for resale. A Windows 95/NT version
of PCx was announced in April 1997, and is available under the same
conditions as the original. (If you want to solve an LP without downloading
a code to your own machine, you can execute PCx remotely through the NEOS
Server.)
A Fortran 77 interior-point code, BPMPD, has been developed by Csaba Mszros
(meszaros@sztaki.hu) at the Computer and Automation Research Institute of
the Hungarian Academy of Sciences. It is available as source code, as a
Windows95/NT executable (which is also extended to solve convex quadratic
problems), and in a DLL version for Windows.
Jacek Gondzio (gondzio@divsun.unige.ch) has made source for his interior
point LP solver HOPDM available at
info.unige.ch/~logilab/software/hopdm.html">http://ecolu-info.unige.ch/~logilab/software/hopdm.html. Additionally,
several papers devoted to HOPDM code are available at this site. It uses a
higher order primal-dual predictor-corrector logarithmic barrier algorithm,
and according to David Gay, it "seems to work well in limited testing. For
example, it happily solves all of the examples in netlib's lp/data
directory." Prof. Gondzio notes that problem size is limited only by
available memory, and on a virtual memory system it has been used to solve
models with hundreds of thousand of constraints and variables. An older
version of the source code is kept in netlib's opt directory:
ftp://netlib.bell-labs.com/netlib/opt/hopdm.shar.Z
Other software of interest:
A web-based service by a group at Berkeley called Interactive Linear
Programming appears to be useful for solving small models that can be
entered by hand. Along similar lines, the NEOS Guide offers a Java-based
Simplex Tool, which demonstrates the workings of the simplex method on small
user-entered problems and is especially useful for educational purposes.
The Systems Analysis Laboratory at Seoul National University offers Linear
Programming software (both Simplex and Barrier) at
http://orly1.snu.ac.kr/Software.html
Will Naylor (naylor@mti.sgi.com) has a collection of software he calls
WNLIB. Routines of interest include
- simplex method for linear programming: contains anti-cycling and numerical
stability hacks. No optimization for sparse matrix.
- transportation problem/assignment problem routine: optimization for sparse
matrix.
Read the INSTALL.txt file for further information. WNLIB also contains
routines pertaining to nonlinear optimization.
The next several suggestions are for public-domain codes that are severely
limited by the algorithm they use (tableau Simplex); they may be OK for
models with (on the order of) 100 variables and constraints, but it's
unlikely they will be satisfactory for larger models. In the words of Matt
Saltzman (mjs@clemson.edu):
The main problems with these codes have to do with scaling, use of
explicit inverses and lack of reinversion, and handling of degeneracy.
Even small problems that are ill-conditioned or degenerate can bring
most of these tableau codes to their knees. Other disadvantages for
larger problems relate to sparsity, pricing, and maintaining the
complete nonbasic portion of the tableau. But for small, dense problems
these difficulties may not be serious enough to prevent tableau codes
from being useful, or even preferable to more "sophisticated" sparse
codes. In any event, use them with care.
* For DOS/PC users, there is an LP and Linear Goal Programming binary
called tslin, at ftp://garbo.uwasa.fi/pc/ts (the current file name is
tslin34.zip, using ZIP compression), or else I suggest contacting Prof.
Salmi at ts@uwasa.fi . For North American users, the garbo server is
mirrored on FTP site wuarchive.wustl.edu, in directory
mirrors/garbo.uwasa.fi.
* Also on the garbo server is a file called lp261.zip, having a
descriptor of "Linear Programming Optimizer by ScanSoft". It consists
of PC binaries, and is evidently some sort of shareware (i.e., not
strictly public domain).
* There is an ACM TOMS routine for LP, #552, available at
ftp://netlib2.cs.utk.edu/toms/552. This routine was designed for
fitting data to linear constraints using an L1 norm, but it uses a
modification of the Simplex Method and could presumably be modified to
satisfy LP purposes.
* There are books that contain source code for the Simplex Method. See
the section on references. You should not expect such code to be
robust. In particular, you can check whether it uses a 2-dimensional
array for the A-matrix; if so, it is surely using the tableau Simplex
Method rather than sparse methods, and Saltzman's comments will apply.
For Macintosh users there is a free package called LinPro that is available
at ftp://ftp.ari.net/MacSciTech/programming/. Some users have reported that
it performs well, while one correspondent informs me he had trouble getting
it to solve any problems at all; perhaps this code is sensitive to memory
size of the machine. It comes with a "large example" of 100 variables, which
gives a hint of its design limits. It seems to be slower than commercial
codes, but that should not be a surprise (or a criticism of a free code).
LinPro has its own input format and does not support MPS format.
Walter C. Riley (73700.776@compuserve.com) writes:
* My shareware program, the R-Tek Scratchpad (rtksp106.zip $15), is
intended for teachers and students. It basically handles problems that
students in an Introduction to Finite Mathematics course might
encounter, including typical small textbook LP problems. Its primary
advantages are that it uses readable math notation, handles fractions,
and allows you to step through the problem to its solution. It is now
available on the net for ftp download at
ftp://ftp.coast.net/SimTel/win3/calc/ or one of its mirror sites.
Stephen F. Gale (sfgale@freenet.calgary.ab.ca) writes:
* Available at http://www.freenet.calgary.ab.ca/~sfgale/simplex.html is a
fairly simple Simplex Solver written for Turbo Pascal 3.0. The original
algorithm is from the book "Some Common BASIC Programs" by Lon Poole
and Mary Borchers (ISBN 0-931988-06-3). However, I revised it
considerably when I converted it to Pascal. I then added Sensitivity
Analysis based on the book "The Operations Research Problem Solver"
(ISBN 0-87891-548-6). I have tested the program on over 30 textbook
problems, but never used it for real life applications. If someone
finds a problem with the program, I would be pleased to correct it. I
would also appreciate knowing how the program was used.
The following suggestions may represent low-cost ways of solving LPs if you
already have certain software available to you.
* All of the most popular spreadsheet programs offer an LP solver as a
feature or add-in.
* A package called QSB (Quantitative Systems for Business, from
Prentice-Hall publishers) has an LP module among its routines.
* If you have access to a commercial math library, such as SAS
(919-677-8000), IMSL (800-222-4675 or 713-784-3131 or
support@houston.vni.com) or NAG (708-971-2337), you may be able to use
an LP routine from there.
* Mathematical systems MATLAB (The Math Works, Inc., (508) 653-1415, see
comment in the NLP FAQ) and MAPLE (Waterloo Maple Software, 450 Phillip
Street, Waterloo, Ontario, Canada N2L 5J2 Phone: (519) 747-2373 Fax:
(519) 747-5284) also have LP solvers. An interface from MATLAB to
lp_solve is available from Jeff Kantor (Jeffrey.Kantor@nd.edu) in
ftp://control.cheg.nd.edu/pub/Kantor/matlab/lp_solve. A MATLAB toolkit
for solving LP models using Interior-Point methods, called LIPSOL is
available at ftp://ftp.math.umbc.edu/pub/zhang/lipsol - check the
documentation in this directory (README.1ST) for more information; the
current version is in subdirectory v0.3. There is an FTP site with
user-contributed .m files to do Simplex located at
ftp://ftp.mathworks.com/pub/contrib/optim/simplex1. There's a Usenet
newsgroup on MATLAB: comp.soft-sys.matlab. If speed matters to you,
then according to a Usenet posting by Pascal Koiran
(koiran@ens-lyon.fr), on randomly generated LP models, MATLAB was an
order of magnitude faster than MAPLE on a 200x20 problem but an order
of magnitude slower than lp_solve on a 50x100 problem. (I don't intend
to get into benchmarking in this document, but I mention these results
just to explain why I choose to focus mostly on special purpose LP
software.)
Commercial codes and modeling systems
If your models prove to be too difficult for free or add-on software to
handle, then you may have to consider acquiring a commercial LP code. Dozens
of such codes are on the market. There are many considerations in selecting
an LP code. Speed is important, but LP is complex enough that different
codes go faster on different models; you won't find a "Consumer Reports"
article to say with certainty which code is THE fastest. I usually suggest
getting benchmark results for your particular type of model if speed is
paramount to you. Benchmarking can also help determine whether a given code
has sufficient numerical stability for your kind of models.
Other questions you should answer: Can you use a stand-alone code, or do you
need a code that can be used as a callable library, or do you require source
code? Do you want the flexibility of a code that runs on many platforms
and/or operating systems, or do you want code that's tuned to your
particular hardware architecture (in which case your hardware vendor may
have suggestions)? Is the choice of algorithm (Simplex, Interior-Point)
important to you? Do you need an interface to a spreadsheet code? Is the
purchase price an overriding concern? If you are at a university, is the
software offered at an academic discount? How much hotline support do you
think you'll need? There is usually a large difference in LP codes, in
performance (speed, numerical stability, adaptability to computer
architectures) and in features, as you climb the price scale.
In the following table is a condensed version of a survey of LP software
that appeared in the June 1992 issue of OR/MS Today (a publication of
INFORMS) and that has subsequently been updated in the October 1995 and
April 1997 issues. Consult the full survey for more detailed information, or
click on the product names to browse their developers' web pages.
The table is in two parts, the first consisting of packages that are
primarily algorithmic codes, and the second containing modeling systems.
Product names are linked to product or developer web sites where known.
Under "Platform" is an indication of common environments in which the code
runs, with the choices being PC-DOS and/or versions of Microsoft Windows
(PC), Macintosh OS (M), and Unix on various computer types (U). For other
possibilities, check the full survey or contact the vendor.
Even more so than usual, I emphasize that you must use this information at
your own risk. I cannot guarantee that every entry is completely correct and
up-to-date, but I will gladly correct any mistakes that are pointed out to
me.
Key to Features: S=Simplex I=Interior-Point or Barrier
Q=Quadratic G=General-Nonlinear
M=MIP N=Network
V=Visualization
Solver
Product Features Platform Phone E-mail address
CPLEX SIMNQ PC M U 702-831-7744 info@cplex.com
C-WHIZ SM PC U 703-412-3201 ketronms@erols.com
FortMP SIMQ PC U 630-971-2337 naginfo@nag.com
+44hossein@unicom.co.uk
1895-256484
HI-PLEX S PC U +44i.maros@ic.ac.uk
171-594-8334
HS/LP SM PC 201-627-1424 info@haverly.com
ILOG
Planner M PC U 415-390-9000 info@ilog.com
LAMPS SM PC U +44info@amsoft.demon.co.uk
181-870-8882
LINDO SMQ PC 312-988-7422 info@lindo.com
LOQO GI PC U 609-258-0876 rvdb@princeton.edu
LPS-867 SM PC U 609-737-6800 info@main.aae.com
LS-XLSOL SM PC 702-831-0300 info@frontsys.com
MINOS SQG PC 415-962-8719 mike@sol-michael.stanford.edu
MINTO M U 404-894-6287 martin.savelsbergh@isye.gatech.edu
MPSIII SMN PC U 703-412-3201 ketronms@erols.com
OSL SIMNQ PC U 914-433-4740 osl@vnet.ibm.com
SAS/OR SMNGQ PC M U 919-677-8000 saseph@unx.sas.com
SCICONIC SM PC U +44msukwt03.gztltm@eds.com
1908-284188
SOPT SIMGQ PC U 908-264-4700 saitech@monmouth.com
XA SM PC M U 818-441-1565 sunsetsw@ix.netcom.com
XPRESS-MP SIMQ PC M 202-887-0296 info@dash.co.uk
+44
1604-858993
Modeling
Product Platform Phone E-mail address
AIMMS PC +31 23-5350935 info@paragon.nl
AMPL PC U 702-322-7600 info@ampl.com
ANALYZE PC 303-796-7830 hgreenbe@carbon.cudenver.edu
DecisionPRO PC 919-859-4101 vginfo@vanguardsw.com
DATAFORM PC U 703-412-3201 ketronms@erols.com
GAMS PC U 202-342-0180 sales@gams.com
LINGO PC U 800-441-2378 info@lindo.com
MathPro PC U 202-887-0296 mathpro@erols.com
MIMI PC U 908-464-8300 info@chesapeake.com
MODLER PC U 303-796-7830 hgreenbe@carbon.cudenver.edu
MPL PC 703-522-7900 info@maximal-usa.com
OMNI PC U 201-627-1424 info@haverly.com
VMP PC U 301-622-4319 j-welch@sundown-vmp.com
What's Best! PC M U 800-441-2378 info@lindo.com
XPRESS-MP PC M 202-887-0296 info@dash.co.uk
+44 1604-858993
Free demos of commercial codes
An increasing number of commercial LP software developers are making demo or
academic versions available for downloading through web sites or as add-ons
to book packages. Typically these versions are limited in the size of
problem they accept or the length of time that they will operate, or are
made available only for "academic use" (mainly research or teaching at
universities). Nevertheless, they have most or all of the features of the
full versions. Most run under several variations of Microsoft Windows on
PCs, and/or certain Unix workstations; check the relevant web pages for
details.
Downloadable free demos include:
* AIMMS with XA and CONOPT
* ANALYZE, MODLER and RANDMOD
* LINDO and What's Best!
* LOQO with a built-in AMPL interface
* MPL with CPLEX
Books that are packaged with demo software include:
* A. Brooke, D. Kendrick and A. Meeraus, GAMS: A Users' Guide, Wadsworth
Publishing Co/Duxbury Press, ISBN 0-894-26215-7.
* R. Fourer, D.M. Gay and B.W. Kernighan, AMPL: A Modeling Language for
Mathematical Programming, Wadsworth Publishing Co/Duxbury Press, ISBN
0-534-50983-5.
* H.J. Greenberg, Modeling by Object-Driven Linear Elemental Relations: A
User's Guide for MODLER, Kluwer Academic Publishers, ISBN
0-792-39323-6.
* L. Schrage, Optimization Modeling with LINDO, LINDO Systems, order
directly from developer.
Many developers are also willing to arrange for you to "borrow" copies of
their full-featured versions for purposes of evaluation. Details vary,
however, so you'll have to check with each vendor whose product you're
interested in.
[ ]
Q3. "Oh, and we also want to solve it as an integer program."
A: Integer LP models are ones whose variables are constrained to take
integer or whole number (as opposed to fractional) values. It may not be
obvious that integer programming is a very much harder problem than ordinary
linear programming, but that is nonetheless the case, in both theory and
practice.
Integer models are known by a variety of names and abbreviations, according
to the generality of the restrictions on their variables. Mixed integer
(MILP or MIP) problems require only some of the variables to take integer
values, whereas pure integer (ILP or IP) problems require all variables to
be integer. Zero-one (or 0-1 or binary) MIPs or IPs restrict their integer
variables to the values zero and one. (The latter are more common than you
might expect, because many kinds of combinatorial and logical restrictions
can be modeled through the use of zero-one variables.)
For the sake of generality, the following disucssion uses the term MIP to
refer to any kind of integer LP problem; the other kinds can be viewed as
special cases. MIP, in turn, is a particular member of the class of
combinatorial or discrete optimization problems. In fact the problem of
determining whether a MIP has an objective value less than a given target is
a member of the class of "NP-complete" problems, all of which are very hard
to solve (at least as far as anyone has been able to tell). Since any
NP-complete problem is reducible to any other, virtually any combinatorial
problem of interest can be attacked in principle by solving some equivalent
MIP. This approach sometimes works well in practice, though it is by no
means infallible.
People are sometimes surprised to learn that MIP problems are solved using
floating point arithmetic. Most available general-purpose large-scale MIP
codes use a procedure called "branch-and-bound" to search for an optimal
integer solution by solving a sequence of related LP "relaxations" that
allow some fractional values. Good codes for MIP distinguish themselves
primarily by solving shorter sequences of LPs, and secondarily by solving
the individual LPs faster. (The similarities between successive LPs in the
"search tree" can be exploited to speed things up considerably.) Even more
so than with regular LP, a costly commercial code may prove its value if
your MIP model is difficult.
Another solution approach known generally as constraint logic programming
(CLP) has drawn increasing interest of late. Having their roots in studies
of logical inference in artificial intelligence, CLP codes typically do not
proceed by solving any LPs. As a result, compared to branch-and-bound they
search "harder" but faster through the tree of potential solutions. Their
greatest advantage, however, lies in their ability to tailor the search to
many constraint forms that can be converted only with difficulty to the form
of an integer program; their greatest success tends to be with "highly
combinatorial" optimization problems such as scheduling, sequencing, and
assignment, where the construction of an equivalent IP would require the
definition of large numbers of zero-one variables. More information and a
list of available codes can be found in the Constraints FAQ (also posted to
the newsgroup comp.constraints).
Whatever your solution technique, you should be prepared to devote far more
computer time and memory to solving a MIP problem than to solving the
corresponding LP relaxation. (Or equivalently, you should be prepared to
solve much smaller MIP problems than LP problems using a given amount of
computer resources.) To further complicate matters, the difficulty of any
particular MIP problem is hard to predict (in advance, at least!). Problems
in no more than a hundred variables can be challenging, while others in tens
of thousands of variables solve readily. The best explanations of why a
particular MIP is difficult often rely on some insight into the system you
are modeling, and even then tend to appear only after a lot of computational
tests have been run. A related observation is that the way you formulate
your model can be as important as the actual choice of solver.
Thus a MIP problem with hundreds of variables (or more) should be approached
with a certain degree of caution and patience. A willingness to experiment
with alternative formulations and with a MIP code's many search options
often pays off in greatly improved performance. In the hardest cases, you
may wish to abandon the goal of a provable optimum; by terminating a MIP
code prematurely, you can often obtain a high-quality solution along with a
provable upper bound on its distance from optimality. A solution whole
objective value is within some fraction of 1% of optimal may be all that is
required for your purposes. (Indeed, it may be an optimal solution. In
contrast to methods for ordinary LP, procedures for MIP may not be able to
prove a solution to be optimal until long after they have found it.)
Once one accepts that large MIP models are not typically solved to a proved
optimal solution, that opens up a broad area of approximate methods,
probabilistic methods and heuristics, as well as modifications to B&B. See
[Balas] which contains a useful heuristic for 0-1 MIP models. See also the
brief discussion of Genetic Algorithms and Simulated Annealing in the
Nonlinear Programming FAQ.
A major exception to this somewhat gloomy outlook is that there are certain
models whose LP solution always turns out to be integer, assuming the input
data is integer to start with. In general these models have a "unimodular"
constraint matrix of some sort, but by far the best-known and most widely
used models of this kind are the so-called pure network flow models. It
turns out that such problems are best solved by specialized routines,
usually based on the simplex method, that are much faster than any
general-purpose LP methods. See the section on Network models for further
information.
Commercial MIP codes are listed with the commercial LP codes and modeling
systems above. The April 1994 issue of OR/MS Today contains a survey of MIP
codes, which largely overlaps the content of the earlier survey on LP:
"Survey: Mixed Integer Programming" by Matthew Saltzman, pp 42-51. The
following are notes on some publicly available codes for MIP problems.
* The public domain code lp_solve, mentioned earlier, accepts MIP models.
* Peter Barth has announced opbdp, an implementation in C++ of an
implicit enumeration algorithm for solving linear 0-1 optimization
problems. The algorithm compares well with commercial linear
programming-based branch-and-bound on a variety of standard 0-1 integer
programming benchmarks. He says that exploiting the logical structure
of a problem, using opbdp, yields good performance on problems where
exploiting the polyhedral structure seems to be inefficient and vice
versa. The package is also available via anonymous ftp at
ftp://ftp.mpi-sb.mpg.de/pub/guide/staff/barth/opbdp/opbdp.tar.Z
along with a Postscript-format technical report (in file mpii952002.ps)
describing the techniques used.
* I have seen mention made of algorithm 333 in the Collected Algorithms
from CACM, though I'd be surprised if it was robust enough to solve
large models. I am not aware of this algorithm being available online
anywhere.
* In [Syslo] is code for 28 algorithms, most of which pertain to some
aspect of Discrete Optimization.
* There is a code called Omega that analyzes systems of linear equations
in integer variables. It does not solve optimization problems, except
in the case that a model reduces completely, but its features could be
useful in analyzing and reducing MIP models. It's available at
ftp.cs.umd.edu:pub/omega (documentation is provided there), or contact
Bill Pugh at pugh@cs.umd.edu.
* Mustafa Akgul (akgul@bilkent.edu.tr) at Bilkent University maintains an
archive in ftp://ftp.bilkent.edu.tr/pub/IEOR/Opt. There is a copy of
lp_solve (though I would recommend using the official source listed in
the previous section), and there is mip386.zip, which is a
zip-compressed code for PC's. He also has a couple of network codes and
various other codes he has picked up. All this is in the further
subdirectories LP, PC, and Network. In addition to the ftp site, there
is gopher (gopher.bilkent.edu.tr), Web (www.bilkent.edu.tr), and
archive-server@bilkent.edu.tr.
* Bob Craig of Lucent Technologies (kat3@ihgp-ebb.ih.lucent.com) has
software written in C, which implements Balas' enumerative algorithm
for solving 0-1 ILP, that he is willing to make available to those who
request it.
[ ]
Q4. "I wrote an optimization code. Where are some test models?"
A: If you want to try out your code on some real-world LP models, there is a
very nice collection of small-to-medium-size ones, with a few that are
rather large, at ftp://netlib2.cs.utk.edu/lp/data, popularly known as the
Netlib collection (although Netlib consists of much more than just LP).
These files (after you uncompress them) are in a format called MPS, which is
described in another section of this document. Note that, when you receive a
model, it may be compressed both with the Unix utility (use `uncompress` if
the file name ends in .Z) AND with an LP-specific program (grab either
emps.f or emps.c at the same time you download the model, then compile/run
the program to reverse the compression).
Also on netlib is a collection of infeasible LP models, located in
ftp://netlib2.cs.utk.edu/lp/infeas.
There is a collection of MIP models, called MIPLIB, housed at Rice
University in http://www.caam.rice.edu/~bixby/miplib/miplib.html. FTP users
can use ftp://ftp.caam.rice.edu/pub/people/bixby/miplib. Or, send an email
message containing "send catalog" to softlib@rice.edu , to get started, if
you can't access the files by other means.
There's a Travelling Salesman Problem library (TSPLIB) in
ftp://softlib.cs.rice.edu/pub/tsplib. (Alternate address:
ftp://elib.zib-berlin.de/pub/mp-testdata/tsp.) A Web version is at
http://www.iwr.uni-heidelberg.de/iwr/comopt/soft/TSPLIB95/TSPLIB.html.
There is a collection of network-flow codes and models at
ftp://dimacs.rutgers.edu/pub/netflow. Another network generator is called
NETGEN and is available at ftp://netlib2.cs.utk.edu/lp/generators.
The commercial modeling language GAMS comes with about 150 test models,
which you might be able to test your code with. AIMMS also comes with some
test models.
There is a collection called MP-TESTDATA available at Konrad-Zuse-Zentrum
fuer Informations-technik Berlin (ZIB) in
ftp://elib.zib-berlin.de/pub/mp-testdata. This directory contains various
subdirectories, each of which has a file named "index" containing further
information. Indexed at this writing are: assign, cluster, lp, ip, matching,
maxflow, mincost, set-parti, steiner-tree, tsp, vehicle-rout, and
generators.
John Beasley maintains the OR-Lib, at ftp://mscmga.ms.ic.ac.uk/pub/, which
contains various optimization test problems. There is an index in
ftp://mscmga.ms.ic.ac.uk/pub/info.txt. WWW access now available at
http://mscmga.ms.ic.ac.uk/. Have a look in the Journal of the Operational
Research Society, Volume 41, Number 11, Pages 1069-72. If you can't access
these resources, send e-mail to umtsk99@vaxa.cc.imperial.ac.uk to get
started. Information about test problems can be obtained by emailing
o.rlibrary@ic.ac.uk with the email message being the file name for the
problem areas you are interested in, or just the word "info".
[ ]
Q5. "What is MPS format?"
A: MPS format was named after an early IBM LP product and has emerged as a
de facto standard ASCII medium among most of the commercial LP codes.
Essentially all commercial LP codes accept this format, but if you are using
public domain software and have MPS files, you may need to write your own
reader routine for this. It's not too hard. See also the comment regarding
the lp_solve code, in another section of this document, for the availability
of an MPS reader.
The main things to know about MPS format are that it is column oriented (as
opposed to entering the model as equations), and everything (variables,
rows, etc.) gets a name. MPS format is described in more detail in
[Murtagh]. A brief description of MPS format is available at
ftp://softlib.cs.rice.edu/pub/miplib
MPS is an old format, so it is set up as though you were using punch cards,
and is not free format. Fields start in column 1, 5, 15, 25, 40 and 50.
Sections of an MPS file are marked by so-called header cards, which are
distinguished by their starting in column 1. Although it is typical to use
upper-case throughout the file (like I said, MPS has long historical roots),
many MPS-readers will accept mixed-case for anything except the header
cards, and some allow mixed-case anywhere. The names that you choose for the
individual entities (constraints or variables) are not important to the
solver; you should pick names that are meaningful to you, or will be easy
for a post-processing code to read.
Here is a little sample model written in MPS format (explained in more
detail below):
NAME TESTPROB
ROWS
N COST
L LIM1
G LIM2
E MYEQN
COLUMNS
XONE COST 1 LIM1 1
XONE LIM2 1
YTWO COST 4 LIM1 1
YTWO MYEQN -1
ZTHREE COST 9 LIM2 1
ZTHREE MYEQN 1
RHS
RHS1 LIM1 5 LIM2 10
RHS1 MYEQN 7
BOUNDS
UP BND1 XONE 4
LO BND1 YTWO -1
UP BND1 YTWO 1
ENDATA
For comparison, here is the same model written out in an equation-oriented
format:
Optimize
COST: XONE + 4 YTWO + 9 ZTHREE
Subject To
LIM1: XONE + YTWO < = 5
LIM2: XONE + ZTHREE > = 10
MYEQN: - YTWO + ZTHREE = 7
Bounds
0 < = XONE < = 4
-1 < = YTWO < = 1
End
Strangely, there is nothing in MPS format that specifies the direction of
optimization. And there really is no standard "default" direction; some LP
codes will maximize if you don't specify otherwise, others will minimize,
and still others put safety first and have no default and require you to
specify it somewhere in a control program or by a calling parameter. If you
have a model formulated for minimization and the code you are using insists
on maximization (or vice versa), it may be easy to convert: just multiply
all the coefficients in your objective function by (-1). The optimal value
of the objective function will then be the negative of the true value, but
the values of the variables themselves will be correct.
The NAME card can have anything you want, starting in column 15. The ROWS
section defines the names of all the constraints; entries in column 2 or 3
are E for equality rows, L for less-than ( <= ) rows, G for greater-than (
>= ) rows, and N for non-constraining rows (the first of which would be
interpreted as the objective function). The order of the rows named in this
section is unimportant.
The largest part of the file is in the COLUMNS section, which is the place
where the entries of the A-matrix are put. All entries for a given column
must be placed consecutively, although within a column the order of the
entries (rows) is irrelevant. Rows not mentioned for a column are implied to
have a coefficient of zero.
The RHS section allows one or more right-hand-side vectors to be defined;
most people don't bother having more than one. In the above example, the
name of the RHS vector is RHS1, and has non-zero values in all 3 of the
constraint rows of the problem. Rows not mentioned in an RHS vector would be
assumed to have a right-hand-side of zero.
The optional BOUNDS section lets you put lower and upper bounds on
individual variables (no * wild cards, unfortunately), instead of having to
define extra rows in the matrix. All the bounds that have a given name in
column 5 are taken together as a set. Variables not mentioned in a given
BOUNDS set are taken to be non-negative (lower bound zero, no upper bound).
A bound of type UP means an upper bound is applied to the variable. A bound
of type LO means a lower bound is applied. A bound type of FX ("fixed")
means that the variable has upper and lower bounds equal to a single value.
A bound type of FR ("free") means the variable has neither lower nor upper
bounds.
There is another optional section called RANGES that I won't go into here.
The final card must be ENDATA, and yes, it is spelled funny.
[ ]
Q6. Topics briefly covered:
Q6.1: "What is a modeling language?"
A: This is any code that creates input for an LP (or MIP, or NLP) code,
using a more flexible input than MPS format. There are no free ones.
Modeling languages can be roughly broken into two classes, column oriented
ones, and equation oriented ones. The former class is older, and includes
such commercial products as OMNI (Haverley Systems) and DATAFORM (Ketron);
they tend to be called "matrix generators" sometimes. Members of the latter
class are GAMS (Scientific Press), LINGO (LINDO Systems), AIMMS (Paragon
Decision Technology) and AMPL (information is in netlib/opt/ampl.info.Z on
the netlib server, or send email to info@ampl.com - see also the WWW home
page for AMPL at http://www.ampl.com/ampl). These products have links to
various solvers, commercial and otherwise.
Broadening the question slightly to cover front-ends in general, it's worth
noting that people frequently use spreadsheet programs as an interface to
some LP solvers.
Q6.2: "How do I diagnose an infeasible LP model?"
A: A linear program is infeasible if the constraints are inconsistent, i.e.,
if no feasible solution can be constructed. It's often difficult to track
down a cause. The cure may even be ambiguous: is it that some demand was set
too high, or a supply set too low? A useful technique is goal programming
(or elastic programming), one variant of which is to include two explicit
slack variables (positive and negative), with huge cost coefficients, in
each constraint. The revised model is guaranteed to have a feasible solution
(at a possibly unbearable cost); constraints that have large slack values in
the "optimal" solution are prime suspects as causes of infeasibility in the
original LP. (Many modelers recommend a an elastic programming philosophy
even if you aren't having trouble achieving feasibility; the idea is that
almost any constraint can be violated, for a great enough price.)
Another approach is to apply auxiliary algorithms that identify constraints
or groups of constraints that can be considered to "cause" the infeasibility
in an LP. A software system called ANALYZE was developed by Harvey Greenberg
to provide computer-assisted analysis, including rule-based intelligence; he
has also compiled a bibliography of more than 400 references on the subject
of model analysis. A system based on the MINOS solver, called MINOS(IIS),
available from John Chinneck (chinneck@sce.carleton.ca), can also be used to
identify a so-called Irreducible Infeasible Subset; the IIS feature is now
available in CPLEX (command "display iis"), OSL, and LINDO (command
"debug"). As a final comment, commercial codes sometimes have other built-in
features to help track infeasibilities.
Q6.3: "I want to know the specific constraints that contradict each other."
A: This may not be a well posed problem. If by this you mean you want to
find the minimal set of constraints that should be removed to restore
feasibility, this can be modeled as an Integer LP (which means, it's
potentially a harder problem than the underlying LP itself). Start with a
Goal Programming approach as outlined above, and introduce some 0-1
variables to turn the slacks off or on. Then minimize on the sum of these
0-1 variables.
John Chinneck (chinneck@sce.carleton.ca) has modified his MINOS(IIS)
extension to find the Irreducible Infeasible Subset as explained in Chinneck
and Dravnieks in the Spring 1991 ORSA Journal on Computing (vol 3, number
2).
Q6.4: "I just want to know whether or not a feasible solution *exists*."
A: From the standpoint of computational complexity, finding out if an LP
model has a feasible solution is essentially as hard as actually finding the
optimal LP solution, within a factor of 2 on average, in terms of effort in
the Simplex Method; plug your problem into a normal LP solver with any
objective function you like, such as c=0. For MIP models, it's also
difficult - if there exists no feasible solution, then you must go through
the entire Branch and Bound procedure (or whatever algorithm you use) to
prove this. There are no shortcuts in general, unless you know something
useful about your model's structure (e.g., if you are solving some form of a
transportation problem, you may be able to assure feasibility by checking
that the sources add up to at least as great a number as the sum of the
destinations).
Q6.5: "I have an LP, except it's got several objective functions."
A: Fundamental to the class of Multiple Criteria models is that there may no
longer be the concept of a unique solution. I am unaware of any public
domain code to approach such problems. I have seen a reference to MATLAB's
Optimization Toolbox.
ADBASE is a package that computes all efficient (i.e., nondominated) extreme
points of a multiple objective linear program. It is available without
charge for research and instructional purposes. If someone has a genuine
need for such a code, they should send a request to: Ralph E. Steuer,
Faculty of Management Science, Brooks Hall, University of Georgia, Athens,
GA 30602-6255.
Other approaches that have worked are:
* Goal Programming (treat the objectives as constraints with costed
slacks), or, almost equivalently, form a composite function from the
given objective functions;
* Pareto preference analysis (essentially brute force examination of all
vertices);
* Put your objective functions in priority order, optimize on one
objective, then change it to a constraint fixed at the optimal value
(perhaps subject to a small tolerance), and repeat with the next
function.
There is a section on this whole topic in [Nemhauser]. [Schrage] has a
chapter devoted to the subject. [Hwang] has also been recommended by a
reader on Usenet. As a final piece of advice, if you can cast your model in
terms of physical realities, or dollars and cents, sometimes the multiple
objectives disappear! 8v)
Q6.6: "I have an LP that has large almost-independent matrix blocks that are
linked by a few constraints. Can I take advantage of this?"
A: Possibly. See section 6.2 in [Nemhauser] for a discussion of
Dantzig-Wolfe decomposition. I am told that the commercial code OSL has
features to assist in doing this. With any other code, you'll have to create
your own framework and then call the LP solver to solve the subproblems. The
folklore is that generally such schemes take a long time to converge so that
they're slower than just solving the model as a whole, although research
continues. For now my advice, unless you are using OSL or your model is so
huge that a good solver can't fit it in memory, is to not bother decomposing
it. It's probably more cost effective to upgrade your solver, if the
algorithm is limiting you, than to invest your time; but I suppose that's an
underlying theme in a lot of my advice. 8v)
Q6.7: "I am looking for an algorithm to compute the convex hull of a finite
number of points in n-dimensional space."
A: There is a program called qhull, available at
ftp://geom.umn.edu/pub/software/qhull.tar.Z. When you uncompress it and
untar it, it will create a directory called qhull which has source code plus
a README file. It uses the "Beneath Beyond" method, described in
[Edelsbrunner].
A code in C called cdd is available at ftp://ifor13.ethz.ch/pub/fukuda/cdd
and is written by Komei Fukuda; download the file named cdd-***.tar.Z, where
*** is the version number (choose the most recent version, which at last
check was 056). It solves the problem stated above, as well as that of
enumerating all vertices. There is also a C++ version at this site (version
073).
A code in C called rs, by David Avis, is available at
ftp://mutt.cs.mcgill.ca/pub/C/README and implements the reverse search
method.
Ken Clarkson has written a program called Hull which is an ANSI C program
that computes the convex hull of a point set in general dimension. The input
is a list of points, and the output is a list of facets of the convex hull
of the points, each facet presented as a list of its vertices. It can be
downloaded from ftp://netlib.bell-labs.com/netlib/voronoi/hull.shar.Z.
There is a directory in ftp://elib.zib-berlin.de/pub/mathprog/polyth that
contains pointers to some tools for such problems.
Nina Amenta has a list of computational geometry software at
http://www.geom.umn.edu/locate/cglist.
Other algorithms for such problems are described in [Swart], [Seidel], and
[Avis]. Such topics are said to be discussed in [Schrijver] (page 224),
[Chvatal] (chapter 18), [Balinski], and [Mattheis] as well. Part of the
method described in [Avis], to enumerate vertices, is implemented in a
Mathematica package called VertexEnum.m at
ftp://cs.sunysb.edu/pub/Combinatorica, in file VE041.Z.
Q6.8: "Are there any parallel LP codes?"
A: The vendors for OSL, CPLEX and XPRESS-MP each have announced parallel
implementations of Branch and Bound solvers for MIP. CPLEX has also
announced parallel implementations of its barrier and dual simplex
algorithms on the Silicon Graphics Power Challenge, and OSL likewise has a
parallel implementation of its barrier method for its SP2 system.
Jeffrey Horn (horn@cs.wisc.edu) has compiled a bibliography of papers
relating to research on parallel B&B. There is an survey article by Gendron
and Crainic in the journal Operations Research, Vol. 42 (1994), No. 6, pp.
1042-1066.
If your particular model is a good candidate for decomposition (see
Decomposition, above) then that form of parallelism could also be very
useful, but you'll have to implement it yourself. Here's what I say to
people who write parallel LP solvers as class projects:
You are probably working with the tableau form of the Simplex method. This
method works well for small models, but it is inefficient for most
real-world models because such models are usually <1% dense. Sparse matrix
methods dominate here. It may well be true that you can get good parallel
speedups with your code, but I'd wager that by the time you get to problems
with 1000 rows, any parallel-dense LP code will be slower than a single-
processor sparse code. And, worse yet, I think it's generally accepted that
no one currently knows how to do a good (i.e., scalable) parallel sparse LP
code. I wouldn't be harping on this point, except that most people's
interest in parallelism is because of the promise of scalability, in which
case large-scale considerations are important. Writing even a
single-processor large-scale LP code is a multi-year project, realistically.
The point is, don't get too enthralled by speedups in your code, unless
there's something to what you are doing that I haven't guessed.
Q6.9: "What software is there for Network models?"
A: In the context of linear programming, the term "network" is most often
associated with the minimum-cost network flow problem. A network for this
problem is viewed as a collection of nodes (or circles or locations) and
arcs (or lines or routes) connecting selected pairs of nodes. Arcs carry a
physical or conceptual flow of some kind, and may be directed (one-way) or
undirected (two-way). Some nodes may be sources (permitting flow to enter
the network) or sinks (permitting flow to leave).
The network linear programming problem is to minimize the (linear) total
cost of flows along all arcs of a network, subject to conservation of flow
at each node, and upper and/or lower bounds on the flow along each arc. This
is a special case of the general linear programming problem. The
transportation problem is an even more special case in which the network is
bipartite: all arcs run from nodes in one subset to the nodes in a disjoint
subset. A variety of other well-known network problems, including shortest
path problems, maximum flow problems, and certain assignment problems, can
also be modeled and solved as network linear programs. Details are presented
in many books on linear programming and operations research.
Network linear programs can be solved 10 to 100 times faster than general
linear programs of the same size, by use of specialized optimization
algorithms. Some commercial LP solvers include a version of the network
simplex method for this purpose. That method has the nice property that, if
it is given integer flow data, it will return optimal flows that are
integral. Integer network LPs can thus be solved efficiently without resort
to complex integer programming software.
Unfortunately, many different network problems of practical interest do not
have a formulation as a network LP. These include network LPs with
additional linear "side constraints" (such as multicommodity flow problems)
as well as problems of network routing and design that have completely
different kinds of constraints. In principle, nearly all of these network
problems can be modeled as integer programs. Some "easy" cases can be solved
much more efficiently by specialized network algorithms, however, while
other "hard" ones are so difficult that they require specialized methods
that may or may not involve some integer programming. Contrary to many
people's intuition, the statement of a hard problem may be only marginally
more complicated than the statement of some easy problem.
A canonical example of a hard network problem is the "traveling salesman"
problem of finding a shortest tour through a network that visits each node
once. A canonical easy problem not obviously equivalent to a linear program
is the "minimum spanning tree" problem to find a least-cost collection of
arcs that connect all the nodes. But if instead you want to connect only
some given subset of nodes (the "Steiner tree" problem) then you are faced
with a hard problem. These and many other network problems are described in
some of the references below.
Software for network optimization is thus in a much more fragmented state
than is general-purpose software for linear programming. Here are some
places to look for software that might serve your purposes:
* MCF, a C implementation of the network simplex method, is available
from Andreas Loebel (loebel@zib.de).
* Several network codes are available by ftp from /pub/netflow at DIMACS
(dimacs.rutgers.edu), including Netflo, the Fortran network simplex
code from [Kennington].
* The minimum-cost network flow code RELAX-IV by Bertsekas and Tseng is
available by ftp from /pub/bertsekas/RELAX at lids.mit.edu.
* Similarly, [Burkard] and [Martello] contain Fortran code for the
Assignment Problem and others.
* There is an ACM TOMS routine, #548, located at
ftp://netlib2.cs.utk.edu/toms/548, that solves the Assignment problem
using the Hungarian Method.
* Ernesto Martins (eqvm@mat.uc.pt) maintains a web page of some public
domain Fortran and C code for various network formulations.
* A package called PPRN, by Jordi Castro (jcastro@etse.urv.es), solves
single or multicommodity network flow problems with linear or nonlinear
objective function and with/without linear side constraints. It is
available by ftp from ftp-eio.upc.es at ftp-eio.upc.es.
Most of these codes are freely available for many purposes, but check their
"readme" files for details.
Q6.10: "What software is there for the Traveling Salesman Problem (TSP)?"
A: TSP is a famously hard problem that has attracted many of the best minds
in the field. Solving for a proved optimum is combinatorial in nature;
methods have been explored both to give proved optimal solutions, and to
give approximate but "good" solutions. To my knowledge, there aren't any
commercial products to solve this problem. Public domain code for the
Asymmetric TSP is available in TOMS routine #750 available at
ftp://netlib2.cs.utk.edu/toms/750; it is documented in [Carpaneto]. For a
bibliography, check the Integer Programming section of [Nemhauser],
particularly the references with the names Groetschel and/or Padberg in
them. A good reference is [Lawler]. Another good one is [Reinelt]. There are
some heuristics for getting a "good" solution in the article by Lin and
Kernighan in Operations Research, Vol 21 (1973), pp 498-516. [Syslo]
contains some algorithms and Pascal code. Numerical Recipes [Press] contains
code that uses Simulated Annealing. [Bentley] is said to contain a
description of how to write a TSP code. Code for a solver can be obtained
via instructions in [Volgenant]. Bob Craig of Lucent Technologies
(kat3@ihgp-ebb.ih.lucent.com) has software written in C, for both exact
solution and heuristics, that he is willing to make available to those who
request it. Likewise, Chad Hurwitz (churritz@cts.com), offers a code called
tsp_solve for heuristic and optimal solution, to those who email him.
Q6.11: "What software is there for the Knapsack Problem?"
A: As with the TSP, I don't know of any commercial solvers for this specific
problem. Any good MIP solver should be able to be used, although any given
instance of this problem could be difficult. Specialized algorithms are said
to be available in [Syslo] and [Martello]. Bob Craig of Lucent Technologies
(kat3@ihgp-ebb.ih.lucent.com) has software written in C, for both exact
solution and heuristics, that he is willing to make available to those who
request it.
Q6.12: "What software is there for Stochastic Programming?"
A: [Thanks to Derek Holmes, dholmes@engin.umich.edu, for this text.] Your
success solving a stochastic program depends greatly on the characteristics
of your problem. The two broad classes of stochastic programming problems
are recourse problems and chance- constrained (or probabilistically
constrained) problems.
Recourse Problems are staged problems wherein one alteranates decisions with
realizations of stochastic data. The objective is to minimize total expected
costs of all decisions. The main sources of code (not necessarily public
domain) depend on how the data is distributed and how many stages (decision
points) are in the problem. For discretely distributed multistage problems,
a good package called MSLiP is available from Gus Gassman
(gassmann@ac.dal.ca, written up in Math. Prog. 47,407-423) Also, for not
huge discretely distributed problems, a deterministic equivalent can be
formed which can be solved with a standard solver. STOPGEN, available via
anonymous FTP from this author is a program which forms deterministic equiv.
MPS files from stopro problems in standard format (Birge, et. al., COAL
newsletter 17). The most recent program for continuously distributed data is
BRAIN, by K. Frauendorfer (frauendorfer@sgcl1.unisg.ch, written up in detail
in the author's monograph ``Stochastic Two-Stage Programming'', Lecture
Notes in Economics & Math. Systems #392 (Springer-Verlag).
CCP problems are not usually staged, and have a constraint of the form Pr(
Ax <= b ) >= alpha. The solvability of CCP problems depends on the
distribution of the data (A &/v b). I don't know of any public domain codes
for CCP probs., but you can get an idea of how to approach the problem by
reading Chapter 5 by Prof. A. Prekopa (prekopa@cancer.rutgers.edu) Y.
Ermoliev, and R. J-B. Wets, eds., Numerical Techniques for Stochastic
Optimization (Series in Comp. Math. 10, Springer-Verlag, 1988).
Both Springer Verlag texts mentioned above are good introductory references
to Stochastic Programming. This list of codes is far from comprehensive, but
should serve as a good starting point.
Q6.13: "I need to do post-optimal analysis."
A: Many commercial LP codes have features to do this. Also called Ranging or
Sensitivity Analysis, it gives information about how the coefficients in the
problem could change without affecting the nature of the solution. Most LP
textbooks, such as [Nemhauser], describe this. Unfortunately, all this
theory applies only to LP.
For a MIP model with both integer and continuous variables, you could get a
limited amount of information by fixing the integer variables at their
optimal values, re-solving the model as an LP, and doing standard
post-optimal analyses on the remaining continuous variables; but this tells
you nothing about the integer variables, which presumably are the ones of
interest. Another MIP approach would be to choose the coefficients of your
model that are of the most interest, and generate "scenarios" using values
within a stated range created by a random number generator. Perhaps five or
ten scenarios would be sufficient; you would solve each of them, and by some
means compare, contrast, or average the answers that are obtained. Noting
patterns in the solutions, for instance, may give you an idea of what
solutions might be most stable. A third approach would be to consider a
goal-programming formulation; perhaps your desire to see post-optimal
analysis is an indication that some important aspect is missing from your
model.
Q6.14: "Do LP codes require a starting vertex?"
A: No. You just have to give an LP code the constraints and the objective
function, and it will construct the vertices for you. Most codes go through
a so-called two phase method, wherein the code first looks for a feasible
solution, and then works on getting an optimal solution. The first phase can
begin anywhere, such as with all the variables at zero (though commercial
codes typically have a so-called "crash" algorithm to pick a better starting
point). So, no, you don't have to give a code a starting point. On the other
hand, it is not uncommon to do so, because it can speed up the solution time
tremendously. Commercial codes usually allow you to do this (they call it a
"basis", though that's a loose usage of a specific linear algebra concept);
free codes generally don't. You'd normally want to bother with a starting
basis only when solving families of related and difficult LP's (i.e., in
some sort of production mode).
Q6.15: "How can I combat cycling in the Simplex algorithm?"
A: Cycling is the condition that occurs when the Simplex method gets "stuck"
and finds itself repeating the same vertices over and over. While this
specific behavior is rather rare in practice, it is quite common for the
algorithm to reach a point where it temporarily stops making forward
progress in terms of improvement in the objective function; this is termed
"stalling", or more loosely known as "degeneracy" since it is caused by one
or more basic variables taking on the value of a lower or upper bound. In
most cases, the algorithm will work through this nest of coincident
vertices, then resume making tangible progress. However, in extreme cases
the degeneracy is so bad that to all intents and purposes it can be
considered cycling.
The simplest answer to the problem of degeneracy/cycling is often to "get a
better optimizer", i.e. one with stronger pricing algorithms, and a better
selection of features. However, obviously that is not always an option
(money!), and even the best LP codes can run into degeneracy on certain
models. Besides, they say it's a poor workman who blames his tools.
So, when one cannot change the optimizer, it's expedient to change the
model. Not drastically, of course, but a little "noise" can usually help to
break the ties that occur during the Simplex method. A procedure that can
work nicely is to add, to the values in the RHS, random values roughly six
orders of magnitude smaller. Depending on your model's formulation, such a
perturbation may not even seriously affect the quality of the solution
values. However, if you want to switch back to the original formulation, the
final solution basis for the perturbed model should be a useful starting
point for a "cleanup" optimization phase. (Depending on the code you are
using, this may take some ingenuity to do, however.)
Another helpful tactic: if your optimization code has more than one solution
algorithm, you can alternate among them. When one algorithm gets stuck,
begin again with another algorithm, using the most recent basis as a
starting point. For instance, alternating between a primal and a dual method
can move the solution away from a nasty point of degeneracy. Using partial
pricing can be a useful tactic against true cycling, as it tends to reorder
the columns. And of course Interior Point algorithms are much less affected
by (though not totally immune to) degeneracy. Unfortunately, the optimizers
richest in alternate algorithms and features also tend to be least prone to
problems with degeneracy in the first place.
[ ]
Q7. "What references and web links are there in this field?"
A: What follows here is an idiosyncratic list, a few books that I like, or
have been recommended on the net, or are recent. I have *not* reviewed them
all.
Regarding the common question of the choice of textbook for a college LP
course, it's difficult to give a blanket answer because of the variety of
topics that can be emphasized: brief overview of algorithms, deeper study of
algorithms, theorems and proofs, complexity theory, efficient linear
algebra, modeling techniques, solution analysis, and so on. A small and
unscientific poll of ORCS-L mailing list readers in 1993 uncovered a
consensus that [Chvatal] was in most ways pretty good, at least for an
algorithmically oriented class; of course, some new candidate texts have
been published in the meantime. For a class in modeling, a book about a
commercial code would be useful (LINDO, AMPL, GAMS were suggested),
especially if the students are going to use such a code; and I have always
had a fondness for the book by [Williams].
General reference
* Nemhauser, Rinnooy Kan, & Todd, eds, Optimization, North-Holland, 1989.
(Very broad-reaching, with large bibliography. Good reference; it's the
place I tend to look first. Expensive, and tough reading for
beginners.)
* Harvey Greenberg has compiled an on-line Mathematical Programming
Glossary.
Books containing source code
* Best and Ritter, Linear Programming: active set analysis and computer
programs, Prentice-Hall, 1985.
* Bertsekas, D.P., Linear Network Optimization: Algorithms and Codes, MIT
Press, 1991.
* Bunday and Garside, Linear Programming in Pascal, Edward Arnold
Publishers, 1987.
* Bunday, Linear Programming in Basic (presumably the same publisher).
* Burkard and Derigs, Springer Verlag Lecture Notes in Math Systems #184
(the Assignment Problem and others).
* Kennington & Helgason, Algorithms for Network Programming, Wiley, 1980.
(A special case of LP; contains Fortran source code.)
* Lau, H.T., A Numerical Library in C for Scientists and Engineers ,
1994, CRC Press. (Contains a section on optimization.)
* Martello and Toth, Knapsack Problems: Algorithms and Computer
Implementations, Wiley, 1990. (Contains Fortran code, comes with a disk
- also covers Assignment Problem.)
* Press, Flannery, Teukolsky & Vetterling, Numerical Recipes, Cambridge,
1986. (Comment: use their LP code with care.)
* Syslo, Deo & Kowalik, Discrete Optimization Algorithms with Pascal
Programs, Prentice-Hall (1983). (Contains code for 28 algorithms such
as Revised Simplex, MIP, networks.)
LP textbooks
* Bazaraa, Jarvis and Sherali. Linear Programming and Network Flows. Grad
level.
* Bertsimas, Dimitris and Tsitsiklis, John, Introduction to Linear
Optimization. Athena Scientific, 1997 (ISBN 1-886529-19-1).
Graduate-level text on linear programming, network flows, and discrete
optimization.
* Chvatal, Linear Programming, Freeman, 1983. Undergrad or grad.
* Daellenbach and Bell, A User's Guide to LP. Good for engineers, but may
be out of print.
* Ecker & Kupferschmid, Introduction to Operations Research.
* Ignizio, J.P. & Cavalier, T.M., Linear Programming, Prentice Hall,
1994. Covers usual LP topics, plus interior point, multi-objective and
heuristic techniques.
* Luenberger, Introduction to Linear and Nonlinear Programming, Addison
Wesley, 1984. Updated version of an old standby.
* Murtagh, B., Advanced Linear Programming, McGraw-Hill, 1981. Good one
after you've read an introductory text.
* Murty, K., Linear and Combinatorial Programming.
* Nash, S., and Sofer, A., Linear and Nonlinear Programming, McGraw-Hill,
1996.
* Nazareth, J.L., Computer Solution of Linear Programs, Oxford University
Press, New York and Oxford, 1987.
* Nering, E.D. & Tucker, A.W., Linear Programs and Related Problems,
Academic Press, 1993.
* Saigal, R., Linear Programming: A Modern Integrated Analysis, Kluwer
Academic Publishers, 1995.
* Schrijver, A., Theory of Linear and Integer Programming, Wiley, 1986.
Advanced.
* Taha, H., Operations Research: An Introduction, 1987.
* Thie, P.R., An Introduction to Linear Programming and Game Theory,
Wiley, 1988.
* Vanderbei, Robert J., Linear Programming: Foundations and Extensions.
Kluwer Academic Publishers, 1996 (ISBN 0-7923-9804-1). Balanced
coverage of simplex and interior-point methods. Source code available
on-line for all algorithms presented.
* Williams, H.P., Model Building in Mathematical Programming, Wiley 1993,
3rd edition. Little on algorithms, but excellent for learning what
makes a good model.
Interior-Point LP methods (descendants of "Karmarkar's algorithm")
* Arbel, Ami, Exploring Interior-Point Linear Programming, MIT Press,
1993. Includes small-scale IBM PC software (binary only).
* Fang and Puthenpura, Linear Optimization and Extensions. (Grad level
textbook, also contains some Simplex and Ellipsoid. I heard mixed
opinions on this one.)
* Lustig, Marsten & Shanno, "Interior Point Methods for Linear
Programming: Computational State of the Art", ORSA Journal on
Computing, Vol. 6, No. 1, Winter 1994, pp. 1-14. Followed by commentary
articles, and a rejoinder by the authors.
* Roos, Terlaky and Vial, Theory and Algorithms for Linear Optimization:
An Interior Point Approach. John Wiley, Chichester, 1997
* Wright, Stephen J., Primal-Dual Interior-Point Methods. SIAM
Publications, 1997. Covers theoretical, practical and computational
aspects of the most important and useful class of interior-point
algorithms. The web page for this book contains current information on
interior-point codes for linear programming, including links to their
web sites.
Presentations of commercially marketed systems (usable as texts for some
classes)
* Bisschop & Entriken, AIMMS: The Modeling System, Paragon Decision
Technology, 1993.
* Brooke, Kendrick & Meeraus, GAMS: A Users' Guide, The Scientific
Press/Duxbury Press, 1988.
* Fourer, Gay & Kernighan, AMPL: A Modeling Language for Mathematical
Programming, The Scientific Press/Duxbury Press, 1992. (Comes with DOS
"student" version including MINOS and CPLEX.)
* Greenberg, H.J., Modeling by Object-Driven Linear Elemental Relations:
A User's Guide for MODLER, Kluwer Academic Publishers, 1993.
* Schrage, L., LINDO: An Optimization Modeling System, The Scientific
Press/Duxbury Press, 1991.
Additional books
* Ahuja, Magnanti & Orlin, Network Flows, Prentice Hall, 1993.
* Beasley, J.E., ed., Advances in Linear and Integer Programming. Oxford
University Press, 1996 (ISBN 0-19-853856-1). Each chapter is a
self-contained essay on one aspect of the subject.
* Bondy & Murty, Graph Theory with Applications.
* Edelsbrunner, Algorithms in Combinatorial Geometry, Springer Verlag,
1987.
* Forsythe, Malcolm & Moler, Computer Methods for Mathematical
Computations, Prentice-Hall.
* Gill, Murray and Wright, Numerical Linear Algebra and Optimization,
Addison-Wesley, 1991.
* Greenberg, H.J., A Computer-Assisted Analysis System for Mathematical
Programming Models and Solutions: A User's Guide for ANALYZE, Kluwer
Academic Publishers, 1993.
* Hwang & Yoon, Multiple Attribute Decision Making : Methods and
Applications, Springer-Verlag, Lecture Notes #186.
* Lawler, Lenstra, et al, The Traveling Salesman Problem, Wiley, 1985.
* More' & Wright, Optimization Software Guide, SIAM Publications, 1993.
See also the NEOS Guide to Optimization Software.
* Murty, Network Programming, Prentice Hall, 1992.
* Papadimitriou & Steiglitz, Combinatorial Optimization. (Also contains a
discussion of complexity of Simplex method.)
* Reeves, C.R., ed., Modern Heuristic Techniques for Combinatorial
Problems, Halsted Press (Wiley), 1993. (Contains chapters on tabu
search, simulated annealing, genetic algorithms, neural nets, and
Lagrangian relaxation.)
* Reinelt, G., The Travelling Salesman: Computational Solutions for TSP
Applications, Springer-Verlag Lecture Notes in Computer Science #840,
1994.
Other publications
* Avis & Fukuda, "A Pivoting Algorithm for Convex Hulls and Vertex
Enumeration of Arrangements and Polyhedra", Discrete and Computational
Geometry, 8 (1992), 295--313.
* Balas, E. and Martin, C., "Pivot And Complement: A Heuristic For 0-1
Programming Problems", Management Science, 1980, Vol 26, pp 86-96.
* Balinski, M.L., "An Algorithm for Finding all Vertices of Convex
Polyhedral Sets", SIAM J. 9, 1, 1961.
* Carpaneto, Dell'amico & Toth, "A Branch-and-bound Algorithm for Large
Scale Asymmetric Travelling Salesman Problems", ACM Transactions on
Mathematical Software (TOMS), December 1995.
* Mattheis and Rubin, "A Survey and Comparison of Methods for Finding All
Vertices of Convex Polyhedral Sets", Mathematics of Operations
Research, vol. 5 no. 2 1980, pp. 167-185.
* Seidel, "Constructing Higher-Dimensional Convex Hulls at Logarithmic
Cost per Face", 1986, 18th ACM STOC, 404--413.
* Smale, Stephen, "On the Average Number of Steps in the Simplex Method
of Linear Programming", Math Programming 27 (1983), 241-262.
* Swart, "Finding the Convex Hull Facet by Facet", Journal of Algorithms,
6 (1985), 17--48.
* Volgenant, A., Symmetric TSPs, European Journal of Operations Research,
49 (1990) 153-154.
On-Line Sources of Papers and Bibliographies
* Michael Trick's Operations Research Page at http://mat.gsia.cmu.edu/
* Optimization Technology Center: home of NEOS, Network-Enabled
Optimization System.
* WORMS (World-Wide-Web for Operations Research and Management Science)
at http://www.maths.mu.oz.au/~worms/
* List of interesting optimization codes in public domain at
http://ucsu.colorado.edu/~xu/software.html. Includes many of the codes
listed here, plus others of interest for specific problem classes.
* Computational Mathematics Archive (London and South East Centre for
High Performance Computing)
http://www.lpac.ac.uk/SEL-HPC/Articles/GeneratedHtml/math.opt.html
* Bibliography of books and survey papers on combinatorial optimization
compiled by Brian Borchers (borchers@nmt.edu),
ftp://archives.math.utk.edu/teaching.materials/bibliography/comb.opt.
* Bibliography of books and papers on Interior-Point methods (taking more
than 400 kilobytes storage with over 1300 entries!?!) in
ftp://netlib2.cs.utk.edu/bib/intbib.bib, compiled by Dr. Eberhard
Kranich (puett@math.uni-wuppertal.de).
* Interior-Point Methods Online (another service of NEOS) contains most
new reports in the area of interior-point methods that have appeared
since December 1994 (over 200 reports as of March 1997). Abstracts for
all reports are available, as are links to postscript source for most
reports . A mailing list is used to notify interested parties whenever
a new report arrives. You can join the list through a web page, or you
can send mail to interior-point-methods-request@mcs.anl.gov containing
the single word subscribe.
* Information related to Semidefinite Programming is at
ftp://orion.uwaterloo.ca/pub/henry/teaching/co769g/readme.html
* An extensive bibliography for stochastic programming has been compiled
by Maarten van der Vlerk at
http://mally.eco.rug.nl/biblio/stoprog.html.
* INFORMS home page is at http://www.informs.org/.
* IMPS Consortium is at http://www-math.cudenver.edu/~hgreenbe/imps.html
On-Line Sources of Optimization Services
The following web sites offer, in some sense, to run your optimization
problem and return a result. Check their home pages for details, which vary
considerably. (Some are intended for nonlinear programming, but are included
here for completeness.)
* DecisionNet. Provides access to "a distributed collection of decision
technologies," including linear programming, "that are made available
for execution over the World Wide Web. These technologies are developed
and maintained locally by their providers. DecisionNet contains
technology metainformation necessary to guide consumers in search,
selection, and execution of these technologies." Facilities for
submitting problems in popular modeling language formats are currently
being tested.
* IBM Optimization Subroutine Library (OSL). Linear and quadratic
programs in MPS format may be submitted by anonymous ftp.
* Internet Enabled HQP Optimization Service. Nonlinear problems in SIF
format may be submitted by e-mail.
* MILP by Dmitry V. Golovashkin. Small-scale mixed-integer programs in a
simple algebraic format are solved through a web form interface.
* Network-Enabled Optimization System (NEOS) Server. Offers access to
about a dozen solvers for a variety of optimization problems, including
linear and nonlinear programming, network and stochastic linear
programming, unconstrained and bound-constrained optimization of
nonlinear functions, and nonlinear complementarity. Linear programs in
MPS format and nonlinear problems in the form of a C or Fortran program
may be submitted by e-mail, Web page, or a high speed, socket-based
Unix interface.
* Numerica. Global nonlinear optimization problems may be submitted in
Numerica's algebraic modeling language, through a web form interface.
[ ]
Q8. "How do I access the Netlib server?"
A: If you have FTP access, you can try "ftp netlib2.cs.utk.edu", using
"anonymous" as the Name, and your email address as the Password. Do a "cd
(dir)" where (dir) is whatever directory was mentioned, and look around,
then do a "get (filename)" on anything that seems interesting. There often
will be a "README" file, which you would want to look at first. Another FTP
site is netlib.bell-labs.com although you will first need to do "cd netlib"
before you can cd to the (dir) you are interested in. Alternatively, you can
reach an e-mail server via "netlib@ornl.gov", to which you can send a
message saying "send index from (dir)"; follow the instructions you receive.
This is a list of sites mirroring the netlib repository:
* Norway netlib@nac.no
* England netlib@ukc.ac.uk
* Germany anonymous@elib.zib-berlin.de
* Taiwan netlib@nchc.edu.tw
* Australia netlib@draci.cs.uow.edu.au
For those who have WWW (Mosaic, etc.) access, one can access Netlib via the
URL http://www.netlib.org. Also, there is, for X window users, a utility
called xnetlib that is available at ftp://netlib2.cs.utk.edu/xnetlib (look
at the "readme" file first).
[ ]
Q9. "Who maintains this FAQ list?"
A: This list was established by John W. Gregory (ashbury@skypoint.com), and
is currently being maintained by Robert Fourer (4er@iems.nwu.edu) and the
Optimization Technology Center.
This article is Copyright 1997 by Robert Fourer and John W. Gregory. It may
be freely redistributed in its entirety provided that this copyright notice
is not removed. It may not be sold for profit or incorporated in commercial
documents without the written permission of the copyright holder. Permission
is expressly granted for this document to be made available for file
transfer from installations offering unrestricted anonymous file transfer on
the Internet.
The material in this document does not reflect any official position taken
by any organization. While all information in this article is believed to be
correct at the time of writing, it is provided "as is" with no warranty
implied.
If you wish to cite this FAQ formally (hey, someone actually asked for
this), you may use:
Fourer, Robert (4er@iems.nwu.edu) and Gregory, John W.
(ashbury@skypoint.com), "Linear Programming FAQ" (1997). World
Wide Web http://www.mcs.anl.gov/home/otc/
faq/linear-programming-faq.html, Usenet sci.answers, anonymous FTP
/pub/usenet/sci.answers/ linear-programming-faq from rtfm.mit.edu.
There's a mail server on rtfm.mit.edu, so if you don't have FTP privileges,
you can send an e-mail message to mail-server@rtfm.mit.edu containing:
send usenet/sci.answers/linear-programming-faq
as the body of the message to receive the latest version (it is posted on
the first working day of each month). This FAQ is cross-posted to
news.answers and sci.op-research.
Suggestions, corrections, topics you'd like to see covered, and additional
material are all solicited. Send email to 4er@iems.nwu.edu.
[ ]
END linear-programming-faq
Last-modified: August 1, 1997