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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. The following are
some of the implementations that are available for downloading. Most are
freely available for many purposes, but check their web pages or "readme"
files for details.

   * ASSCT, an implementation of the Hungarian Method for the Assignment
     problem (#548 from Collected Algorithms of the ACM).

   * GIDEN, an interactive graphical environment for a variety of network
     problems and algorithms, available as a Java application or as an
     applet that can be executed through any Java-enabled Web browser.
     Further information is available by writing to giden@iems.nwu.edu.

   * MCF, a C implementation of the network simplex method (from Andreas
     Loebel, loebel@zib.de).

   * Netflo, the Fortran network simplex code from [Kennington], and several
     codes for maximum matching and maximum flow problems (from DIMACS,
     help@dimacs.rutgers.edu)

   * PPRN, for single or multicommodity network flow problems having a
     linear or nonlinear objective function, optionally with linear side
     constraints, by Jordi Castro (jcastro@etse.urv.es)

   * RELAX-IV for minimum-cost network flows (by Dimitri Bertsekas,
     bertsekas@lids.mit.edu and Paul Tseng, tseng@math.washington.edu); also
     a C++ version of the RELAX-IV algorithm (at the Department of Computer
     Science, University of Pisa, frangio@di.unipi.it)

The following indexes may also be useful:

   * Network optimization codes in Fortran 77 and in C, compiled by Ernesto
     Martins (eqvm@mat.uc.pt)

   * The network optimization library, including codes for assignment,
     shortest path, minimum-cost flow, and maximum flow/minimum cut, by
     Andrew Goldberg (avg@research.nj.nec.com).

   * Optimization routines for networks and graphs in the listing of
     public-domain optimization codes maintained by Jiefeng Xu
     (Jiefeng.Xu@Colorado.edu).

   * Network optimization listings from the NEOS Guide.

Fortran code for the Assignment Problem and others can also be copied
from[Burkard] and from [Martello].

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.

   * GIDEN. An interactive graphical environment for a variety of network
     optimization problems and algorithms. It is written in Java, so you can
     try it out through any Java-enabled Web browser.

   * 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 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 sending e-mail, by submitting
     URLs through a Web page, or via a high-speed socket-based Unix
     interface. Linear and nonlinear programs in the AMPL modeling language
     can also be sent to some of the solvers, by e-mail or URL.

   * NIMBUS. A multiobjective optimization system that accepts algebraic
     problem specifications through a series of Web forms.

   * 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.

[ ]

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