Danny Amor's Online Library/Danny Amor's Online Library - Volume 1.iso/html/faqs/faq/nonlinear-programming-faq

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Subject: Nonlinear Programming FAQ
Newsgroups: news.answers,sci.answers,sci.op-research
From: [e-mail] (John Gregory)
Date: 1 Nov 94 09:03:01 CST

Posted-By: auto-faq 2.4
Archive-name: nonlinear-programming-faq
Last-modified: November 1, 1994

     Nonlinear Programming - Frequently Asked Questions List

Posted monthly to Usenet newsgroup sci.op-research
World Wide Web version:
  ftp://ftp.cray.com/pub/FAQs/nonlinear-programming-faq.html
Plain-text version:
  ftp://rtfm.mit.edu/pub/usenet/sci.answers/nonlinear-programming-faq

Date of this version: November 1, 1994

"Numeric stability is probably not all that important when you're
guessing." -- Anonymous

 o Q1. "What is Nonlinear Programming?"
 o Q2. "What software is there for nonlinear optimization?"
 o Q3. "I wrote an optimization code. Where are some test models?"
 o Q4. "What references are there in this field?"
 o Q5. "How do I access the Netlib server?
 o Q6. "Who maintains this FAQ list?"

See also the related Linear Programming FAQ .


Q1. "What is Nonlinear Programming?"
+++++++++++++++++++++++++++++++++++++

A: A Nonlinear Program (NLP) is a problem that can be put into
the form

    minimize   F(x)
    subject to g (x)  = 0     for i=1,...,m1       where m1 >= 0
                i
               h (x) >= 0    for j=m1+1,...,m      where m >= m1
                j

That is, there is one scalar-valued function F, of several variables
(x here is a vector), that we seek to minimize subject (perhaps) to
one or more other such functions that serve to limit or define the
values of these variables. F is called the "objective function", while
the various other functions are called the "constraints". (If
maximization is sought, it is trivial to do so, by multiplying F by
-1.)

Because NLP is a difficult field, researchers have identified special
cases for study. A particularly well studied case is the one where all
the constraints g and h are linear. The name for such a problem,
unsurprisingly, is "linearly constrained optimization". If, as well,
the objective function is quadratic at most, this problem is called
Quadratic Programming (QP). A further special case of great
importance is where the objective function is entirely linear; this is
called Linear Programming and is discussed in a separate FAQ list.
Another important special case, called unconstrained optimization,
is where there are no constraints at all.

One of the greatest challenges in NLP is that some problems
exhibit "local optima"; that is, spurious solutions that merely satisfy
the requirements on the derivatives of the functions. Think of a
near-sighted mountain climber in a terrain with multiple peaks,
and you'll see the difficulty posed for an algorithm that tries to
move from point to point only by climbing uphill. Algorithms that
propose to overcome this difficulty are termed "Global
Optimization".

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 "NLP 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.


Q2. "What software is there for nonlinear optimization?"
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++

A: It's unrealistic to expect to find one general NLP code that's
going to work for every kind of nonlinear model. Instead, you
should try to select a code that fits the problem you are solving. If
your problem doesn't fit in any category except "general", or if you
insist on a globally optimal solution (except when there is no
chance of encountering multiple local optima), you should be
prepared to have to use a method that boils down to exhaustive
search, i.e., you have an intractable problem.

Several of the commercial LP codes referenced in the LP FAQ
have specialized routines, particularly QP. The ones that I know of
that have some form of QP are: LINDO, KORBX, LOQO,
MPS-III, OSL, and PC-PROG. Of course, you don't generally get
source code when you license one of these products; but many of
them can be licensed as a callable library of solver routines. Many
general nonlinear problems can be solved (or at least confronted)
by application of a sequence of LP or QP approximations.

There are ACM TOMS routines for QP, #559 and #587, available
in ftp://netlib2.cs.utk.edu/toms/559 and
ftp://netlib2.cs.utk.edu/toms/587

There is a directory on Netlib, ftp://netlib2.cs.utk.edu/opt,
containing a collection of optimization routines. The last time I
checked, I saw

 o "praxis" (unconstrained optimization, without requiring
   derivatives)
 o "tn" (Newton method for unconstrained or simple-bound
   optimization)
 o "ve08" (optimization of unconstrained separable function).
 o "simann" (unconstrained optimization using Simulated
   Annealing)
 o "subplex"(unconstrained optimization, general multivariate
   functions)
 o "donlp" (differentiable nonlinear optimization, dense linear
   algebra)
 o "hooke" (Hooke and Jeeves method)

A package called conmin (unrelated to the one by Vanderplaats
and Associates), is available at
ftp://anusf.anu.edu.au/mld900/conmin. Any comments should be
sent to Murray Dow at [e-mail]. The author
states that it is reliable but not state of the art; surpassed, for
instance, by FSQP.

NSWC Library of Mathematical Subroutines has a subroutine to
minimize a function of n variables (OPTF) and a subroutine to
solve a system of nonlinear equations (HBRD). Such routines are
also available in NMS library [Kahaner].

For nonlinear optimization problems with both continuous and
binary variables (MINLP), there is a code called DICOPT++,
available commercially from GAMS Development Corp. Contact
[e-mail] for more information.

Microsoft Excel 4.0 and above has a function called "Solver", based
on GRG2. This product runs on PC and Macintoshes. The
attraction of this approach is that models can be built using the
spreadsheet. I am told that this function can handle 200
independent variables and 500 constraints.

For difficult problems like Global Optimization, methods like
Genetic Algorithms and Simulated Annealing have been studied
heavily. I'm not well-versed in any of these topics, and I have been
assured of contradictory things by different experts. A particular
point of controversy is whether there is a proof of optimality for
practical variants of such algorithms for Global Optimization
problems, and I take no particular stand on the issue (since for
difficult problems such a proof may be of academic interest only).
Even moreso than usual, I say "let the user beware" when it comes
to code. There's a (compressed) Postscript file available at
ftp://beethoven.cs.colostate.edu/pub/TechReports/1993/tr-103.ps.Z,
containing a forty-page introduction to the topic of GA. The
Usenet newsgroup on GA, comp.ai.genetic, has a FAQ on the topic,
otherwise known as "The Hitch-Hiker's Guide to Evolutionary
Computation", available at
ftp://rtfm.mit.edu/pub/usenet/news.answers/ai-faq/genetic. Genetic
Algorithm code can be obtained at ftp://cs.ucsd.edu/pub/GAucsd.
Simulated Annealing code for NLP and other problems is available
at ftp://ftp.alumni.caltech.edu/pub/ingber - contact Lester Ingber
([e-mail]) for more info. A code called SDSC
EBSA (Ensemble Based Simulated Annealing) is available at
ftp://ftp.sdsc.edu/pub/sdsc/math/Ebsa, or contact Richard Frost
([e-mail]). And there is the simann code available on Netlib,
mentioned above. For other ideas on Global Optimization, you may
want to consult one of the books given in the section on references ,
such as [Nemhauser] or one of the ones with "Global" in its title.
There is also the Journal of Global Optimization, published by
Kluwer.

Here is a summary of other NLP codes mentioned in newsgroups in
the past few years, sorted alphabetically. Perhaps someone will
volunteer to organize these references more usefully.

 o Amoeba - Numerical Recipes
 o Brent's Method - Numerical Recipes
 o CONMIN - Vanderplaats, Miura & Associates, Colorado
   Springs, Colorado, 719-527-2691.
 o CONOPT - large-scale GRG code, by Arne Drud.
   Available with GAMS, AIMMS, or AMPL (modeling
   languages - see LP FAQ) or standalone.
 o DFPMIN - Numerical Recipes (Davidon-Fletcher-Powell)
 o Eureka - Borland Software (for IBM PC class of machines)
 o FSQP/CFSQP (Fortran/C) - Contact Andre Tits,
   [e-mail]. Free of charge to academic users.
   Solves large-scale general nonlinear constrained problems,
   including constrained minimax problems. CFSQP (C code)
   includes a scheme to efficently handle problems with many
   constraints (e.g., discretized semi-infinite problems).
 o GENOCOP - Solves linearly constrained problems via a
   Genetic algorithm, available at ftp://unccsun.uncc.edu.
   Author: Zbigniew Michalewicz, [e-mail].
 o GINO - LINDO Systems, Chicago, IL
 o GRG2 - Leon Lasdon, University of Texas, Austin TX
 o Harwell Library routine VF02AD (or is it VF04?).
 o Hooke and Jeeves algorithm - see reference below. A
   version is available at ftp://netlib2.cs.utk.edu/opt/hooke.c,
   and may be useful because it handles nondifferentiable
   and/or discontinuous functions.
 o IMSL routine UMINF and UMIDH.
 o LANCELOT - large-scale NLP. See the book by Conn et
   al. in the references section. For peaceful purposes only.
 o LSSOL - Stanford Business Software Inc. (See MINOS)
   This code does convex (positive semi-definite) QP. Is the
   QP solver used in current versions of NPSOL.
 o MATLAB Optimization Toolbox - The Mathworks, Inc.
   508-653-1415. Handles various nonlinear optimization
   problems. Data sheet available in postscript format at
   ftp://ftp.mathworks.com/pub/product-info/optimization.ps
   Email address: [e-mail] .
 o MINOS - Stanford Business Software Inc., 415-962-8719.
   Mailing address: 2672 Bayshore Parkway, Suite 304,
   Mountain View, CA 94043. Email:
   [e-mail]. This large-scale code is
   often used by researchers as a "benchmark" for others to
   compare with.
 o MINPACK I and II - Contact Jorge More',
   [e-mail], or check ftp://netlib2.cs.utk.edu/minpack.
   Solves dense nonlinear least-squares problems.
 o NAG Library routine E04UCF (essentially the same as
   NPSOL).
 o Nelder and Mead's method - Numerical Recipes, also
   Barabino.
 o NOVA - DOT Products, Houston TX
 o NPSOL - Stanford Business Software Inc. (See MINOS)
 o QLD - Contact: [e-mail].
   Solves Quadratic Programming and other nonlinear
   problems.
 o QPSOL - see LSSOL.
 o SLATEC - Quadratic solvers dbocls, dlsei, and other
   routines. National Energy Software Center, 9700 Cass Ave.,
   Argonne, Illinois 60439. Also available at
   ftp://netlib2.cs.utk.edu/slatec

A book that became available in November 1993 is the
"Optimization Software Guide," by Jorge More' and Stephen
Wright, from SIAM Books. Call 1-800-447-7426 to order ($24.50,
less ten percent if you are a SIAM member). It contains references
to 75 available software packages, and goes into more detail than is
possible in this FAQ.

I would be extremely interested in hearing of people's experiences
with the codes they learn about from reading this FAQ. (Note, I'm
looking for more-or-less independent confirmation or denial of the
practicality of codes.)


Q3. "I wrote an optimization code. Where are some test models?"
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

A: There are various test sets for NLP. Among those I've seen
mentioned are:

 o A. Corana et al, "Minimizing Multimodal Functions of
   Continuous Variables with the Simulated Annealing
   Algorithm," ACM Transactions on Mathematical Software,
   Vol. 13, No. 3, Sept 1987, pp. 262-280. (Difficult
   unconstrained nonlinear models)
 o C.A. Floudas and P.M. Pardalos, A Collection of Test
   Problems for Constrained Global Optimization Algorithms,
   Springer-Verlag, Lecture Notes in Computer Science 455
   (1990).
 o W.W. Hager, P.M. Pardalos, I.M. Roussos, and H.D.
   Sahinoglou, Active Constraints, Indefinite Quadratic
   Programming, and Test Problems, Journal of Optimization
   Theory and Applications Vol. 68, No. 3 (1991), pp. 499-511.
 o J. Hasselberg, P.M. Pardalos and G. Vairaktarakis, Test case
   generators and computational results for the maximum
   clique problem, Journal of Global Optimization 3 (1993), pp.
   463-482.
 o B. Khoury, P.M. Pardalos and D.-Z. Du, A test problem
   generator for the steiner problem in graphs, ACM
   Transactions on Mathematical Software, Vol. 19, No. 4
   (1993), pp. 509-522.
 o Y. Li and P.M. Pardalos, Generating quadratic assignment
   test problems with known optimal permutations,
   Computational Optimization and Applications Vol. 1, No. 2
   (1992), pp. 163-184.
 o P. Pardalos, "Generation of Large-Scale Quadratic
   Programs", ACM Transactions on Mathematical Software,
   Vol. 13, No. 2, p. 133.
 o P.M. Pardalos, Construction of test problems in quadratic
   bivalent programming, ACM Transactions on Mathematical
   Software, Vol. 17, No. 1 (1991), pp. 74-87.
 o P.M. Pardalos, Generation of large-scale quadratic
   programs for use as global optimization test problems, ACM
   Transactions on Mathematical Software, Vol. 13, No. 2
   (1987), pp. 133-137.
 o F. Schoen, "A Wide Class of Test Functions for Global
   Optimization", J. of Global Optimization, 3, 133-137, 1993,
   with C source code available at
   ftp://ghost.dsi.unimi.it/pub/schoen.
 o publications (referenced in another section of this list) by
   Schittkowski; Hock & Schittkowski; Torn & Zilinskas.

Some of the other publications listed in the references section also
may contain problems that you could use to test a code.

A package called CUTE (Constrained and Unconstrained Testing
Environment) is a set of Fortran subroutines, system tools and test
problems in the area of nonlinear optimization and nonlinear
equations, available at ftp://camelot.cc.rl.ac.uk/pub/cute. or at
ftp://thales.math.fundp.ac.be/cute. A LaTex formatted manuscript is
included in the distribution file. Download the README file first
and follow the directions contained therein. Questions should be
directed toward any of the package's authors:

 o Ingrid Bongartz [e-mail]
 o Andy Conn [e-mail]
 o Nick Gould [e-mail]
 o Philippe Toint [e-mail]

John Beasley has posted information on his OR-Lib, which
contains various optimization test problems. Send e-mail to
[e-mail] to get started. Or have a look in
the Journal of the Operational Research Society, Volume 41,
Number 11, Pages 1069-72. Available at
ftp://mscmga.ms.ic.ac.uk/pub. The only nonlinear models in this
collection at this writing are Quadratic Assignment problems.

The modeling language GAMS comes with about 150 test models,
which you might be able to test your code with. The models are in
the public domain according to the vendor, although you need
access to a GAMS system if you want to run them without
modifying the files. The modeling system AIMMS also comes with
a number of test models.

In the journal Mathematical Programming, Volume 61 (1993)
Number 2, there is an article by Calamai et al. that discusses how
to generate QP test models. It gives what seems a very full
bibliography of earlier articles on this topic. The author offers at
the end of the article to send a Fortran code that generates QP
models, if you send email to [e-mail].

The paper "An evaluation of the Sniffer Global Optimization
Algorithm Using Standard Test Functions", Roger A.R. Butler and
Edward E. Slaminka, J. Comp. Physics, 99, 28-32, (1992) mentions
the following reference containing 7 functions that were intended
to thwart global minimization algorithms: "Towards Global
Optimization 2", L.C.W. Dixon and G.P. Szego, North-Holland,
1978. [from Dean Schulze - [e-mail]]


Q4. "What references 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. I have *not* reviewed them
all. I have marked with an arrow ("->") books that received
positive mention in an informal poll on Usenet, regarding good
textbooks for a course on optimization.

General reference

 o 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.)

Other publications (can someone help classify these more usefully?)

 o Barabino, et al, "A Study on the Performances of Simplex
   Methods for Function Minimization," 1980 Proceedings of
   the IEEE International Conference on Circuits and
   Computers, (ICCC 1980), pp. 1150-1153.
 o -> Bazaraa, Shetty, & Sherali, Nonlinear Programming,
   Theory & Applications, Wiley, 1994.
 o Coleman & Li, Large Scale Numerical Optimization, SIAM
   Books.
 o Conn, A.R., et al., "LANCELOT: A code for large-scale,
   constrained, NLP", Springer series in computational
   mathematics, 1992.
 o Dennis & Schnabel, Numerical Methods for Unconstrained
   Optimization and Nonlinear Equations, Prentice Hall, 1983.
 o Fiacco & McCormick, Sequential Unconstrained
   Minimization Techniques, SIAM Books. (An old standby,
   given new life by the interior point LP methods.)
 o Fletcher, R., Practical Methods of Optimization, Wiley,
   1987. (Good reference for Quadratic Programming, among
   other things.)
 o Floudas & Pardalos, Recent Advances in Global
   Optimization, Princeton University Press, 1992.
 o Gill, Murray & Wright, Practical Optimization, Academic
   Press, 1981. (An instant NLP classic when it was
   published.)
 o Himmelblau, Applied Nonlinear Programming,
   McGraw-Hill, 1972. (Contains some famous test problems.)
 o Hock & Schittkowski, Test Examples for Nonlinear
   Programming Codes, Springer-Verlag, 1981.
 o Hooke & Jeeves, "Direct Search Solution of Numerical and
   Statistical Problems", Journal of the ACM, Vol.8 pp.
   212-229, April 1961.
 o Horst and Tuy, Global Optimization, Springer-Verlag,
   1993.
 o Kahaner, Moler & Nash, Numerical Methods and Software,
   Prentice- Hall.
 o -> Luenberger, Introduction to Linear and Nonlinear
   Programming, Addison Wesley, 1984. (Updated version of
   an old standby.)
 o More', "Numerical Solution of Bound Constrained
   Problems", in Computational Techniques & Applications,
   CTAC-87, Noye & Fletcher, eds, North-Holland, 29-37,
   1988.
 o More' & Toraldo, Algorithms for Bound Constrained
   Quadratic Programming Problems, Numerische Mathematik
   55, 377-400, 1989.
 o More' & Wright, "Optimization Software Guide", SIAM,
   1993.
 o Nocedal, J., summary of algorithms for unconstrained
   optimization in "Acta Numerica 1992".
 o Pardalos & Wolkowicz, eds., Quadratic Assignment and
   Related Problems, American Mathematical Society,
   DIMACS series in discrete mathematics, 1994.
 o Powell, M.J.D., "A Fast Algorithm for Nonlinearly
   Constrained Optimization Calculations", Springer-Verlag
   Lecture Notes in Mathematics, vol. 630, pp. 144-157.
 o Press, Flannery, Teukolsky & Vetterling, Numerical Recipes,
   Cambridge, 1986.
 o Schittkowski, Nonlinear Programming Codes,
   Springer-Verlag, 1980.
 o Schittkowski, More Test Examples for Nonlinear
   Programming Codes, Lecture Notes in Economics and
   Math. Systems 282, Springer 1987.
 o Torn & Zilinskas, Global Optimization, Springer-Verlag,
   1989.
 o Wismer & Chattergy, Introduction to Nonlinear
   Optimization, North-Holland, 1978. (Undergrad text)
 o Wright, M., "Interior methods for constrained optimization",
   Acta Mathematica, Cambridge University Press, 1992.
   (Survey article.)

Simulated Annealing & Genetic Algorithms

 o Davis, L. (ed.), Genetic Algorithms and Simulated
   Annealing, Morgan Kaufmann, 1989.
 o De Jong, "Genetic algorithms are NOT function optimizers"
   in Foundations of Genetic Algorithms: Proceedings 24-29
   July 1992, D. Whitley (ed.) Morgan Kaufman
 o Goldberg, D., "Genetic Algorithms in Search, Optimization,
   and Machine Learning", Addison-Wesley, 1989.
 o Ingber "Very fast simulated re-annealing" Mathematical
   and Computer Modeling, 12(8) 1989, 967-973
 o Kirkpatrick, Gelatt & Vecchi, Optimization by Simulated
   Annealing, Science, 220 (4598) 671-680, 1983.
 o Michalewicz et al., article in volume 3(4) 1991 of the ORSA
   Journal on Computing.
 o Michalewicz, Z., "Genetic Algorithms + Data Structures =
   Evolution Programs", Springer Verlag, 1992.
 o Reeves, C.R., ed., Modern Heuristic Techniques for
   Combinatorial Problems, Halsted Press (Wiley). (Contains
   chapters on tabu search, simulated annealing, genetic
   algorithms, neural nets, and Lagrangean relaxation.)

On-Line Papers

 o Computational Mathematics Archive (London and South
   East Centre for High Performance Computing)
   http://www.lpac.qmw.ac.uk/SEL-HPC/Articles/GeneratedHtml/math.opt.html


Q5. "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.att.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 "[e-mail]", to which you can
send a message saying "send index from (dir)"; follow the
instructions you receive. This is the list of sites mirroring the
netlib repository:

 o Norway [e-mail]
 o England [e-mail]
 o Germany [e-mail]
 o Taiwan [e-mail]
 o Australia [e-mail]

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).


Q6. "Who maintains this FAQ list?"
+++++++++++++++++++++++++++++++++++

A:  John W. Gregory     [e-mail]         612-683-3673
    Applications Dept.  Cray Research, Inc.  Eagan, MN 55121 USA

This article is Copyright 1994 by 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 Cray Research, Inc. 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
me this), you may use:

  Gregory, John W. ([e-mail]) "Nonlinear Programming FAQ",
  (1994) Usenet sci.answers.  Available via anonymous FTP
  from rtfm.mit.edu in
  /pub/usenet/sci.answers/nonlinear-programming-faq

There's a mail server on that machine, so if you don't have FTP
privileges, you can send an e-mail message to
[e-mail] containing:

    send usenet/sci.answers/nonlinear-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. If you have access to a World
Wide Web server (Mosaic, Lynx, etc.), you can use
ftp://rtfm.mit.edu/pub/usenet/sci.answers/nonlinear-programming-faq.

ftp://rtfm.mit.edu/pub/usenet/news.answers/nonlinear-programming-faq.

ftp://rtfm.mit.edu/pub/usenet/sci.op-research/nonlinear-programming-faq.

In compiling this information, I have drawn on my own knowledge
of the field, plus information from contributors to Usenet groups
and the ORCS-L mailing list. I give my thanks to all those who
have offered advice and support. I've tried to keep my own biases
(primarily, toward the high end of computing) from dominating
what I write here, and other viewpoints that I've missed are
welcome. Suggestions, corrections, topics you'd like to see covered,
and additional material, are all solicited.


END nonlinear-programming-faq