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[ ]

        Nonlinear 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:
Plain-text version:

Date of this version: December 1, 1997

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

See also the following pages
pertaining to mathematical programming and optimization modeling:

   * The related Linear Programming FAQ.
   * The NEOS Guide to optimization models and software.
   * The Decision Tree for Optimization Software by H.D. Mittelmann and P.
   * Jiefeng Xu's List of Interesting Optimization Codes in the Public
   * Software for Optimization: A Buyer's Guide by Robert Fourer.
   * Harvey Greenberg's Mathematical Programming Glossary.

[ ]

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 gi(x)  = 0    for i = 1, ..., m1       where m1 >= 0
               hj(x) >= 0    for j = m1+1, ..., m     where m >= m1

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 (LP) 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

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.

If you simply want to try solving a particular model, consider the
Optimization Technology Center at
The centerpiece of this ambitious project is NEOS, the Network-Enhanced
Optimization System, consisting of a library of optimization software, a
facility to use this library over the network at, and a guide to more
information about the software packages. Linear and nonlinear models are
covered. Capabilities and access modes are still being enhanced - this is an
ongoing process.

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 and

There is a directory on Netlib,, containing a
collection of optimization routines. The last time I checked, I saw

   * "praxis" (unconstrained optimization, without requiring derivatives)
   * "tn" (Newton method for unconstrained or simple-bound optimization)
   * "ve08" (optimization of unconstrained separable function).
   * "simann" (unconstrained optimization using Simulated Annealing)
   * "subplex"(unconstrained optimization, general multivariate functions)
   * "donlp2" (differentiable nonlinear optimization, dense linear algebra)
   * "hooke" (Hooke and Jeeves method)

A newer version of the "donlp2" code, mentioned above, can be found at This is P. Spellucci's implementation of
a SQP method for general nonlinear optimization problems including nonlinear
equality and inequality constraints (generally referred to as the NLP
problem). It is freely available for non-commercial and research use, and
includes a number of nontrivial examples. There are four versions:

   * donlp2.tar.gz, donlp2_d.tar.gz (f77, exact & numerical gradients
   * donlp2_c.tar.gz, donlp2_d_c.tar.gz (f2c-converted C-versions of the

A package for large optimization problems (with only simple bounds for
constraints), L-BFGS-B, implements a limited memory BFGS algorithm. The user
must supply the gradient g of f, but knowledge about the Hessian matrix is
not required. This program is an extension of algorithm L-BFGS (Harwell
routine VA15) which can handle only unconstrained problems. Both codes can
be obtained via anonymous ftp at and

A package called conmin (unrelated to the one by Vanderplaats and
Associates), is available at or Any comments should be sent to
Murray Dow at The author states that it is reliable
but not state of the art; surpassed, for instance, by FSQP.

Will Naylor ( has a collection of software he calls
WNLIB. Routines of interest include
- unconstrained non-linear optimization routines: implementation of
conjugate-gradient and conjugate-directions algorithms.
- constrained non-linear optimization routines: based on conjugate-gradient
algorithm with penalties.
- 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
- general simulated annealing routine
These routines can be obtained by anonymous ftp from

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

SolvOpt, by Alexei Kuntsevich ( and Franz Kappel
(, is designed for local optimization of
nonsmooth nonlinear problems. Free source code is available in C and
Fortran, and also as M-functions for use with MATLAB. Further information is
provided by a manual that is also available for downloading.

For nonlinear optimization problems with both continuous and binary
variables (MINLP), there is a code called DICOPT++, available commercially
from GAMS Development Corp. Contact for more information.
(There is a survey article, "Constrained Nonllinear 0-1 Programming" by
Hansen, Jaumard, and Mathon, in the ORSA Journal on Computing, 5, 2, Spring

While they are not NLP solvers, per se, attention should be given to
modeling languages like GAMS, AIMMS and AMPL. See the Linear Programming FAQ
for details on contacting the vendors of these and other modeling language

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. Quattro also has a
solver based on GRG2.

A package that uses Microsoft Excel as its input mechanism is Magestic, a
non-linear least squares minimization program. You can contact the vendor
at: Logix Consulting, Inc., 11408 Audelia, Ste 4944, Dallas, TX 752431,
1-800-900-5541 or 214-918-0700.

O-Matrix for Windows includes several non-linear optimization tools. You can
contact the vendor at: Harmonic Software Inc., 12223 Dayton Avenue North,
Seattle WA 98133, 1-800-895-4546, 206-367-8742.

Information for obtaining ACCPM, which implements an analytic center cutting
plane method for convex optimization problems, is available at

Semidefinite Programming is a generalization of linear programming to the
space of block diagonal, symmetric, positive semidefinite matrices. Interest
in this topic, which has numerous engineering applications, has been greatly
stimulated by the extension of interior-point methods from linear
programming to the semidefinite case. Several groups of researchers are
developing interior-point codes, such as:

   * CSDP, a subroutine library available in C or Fortran source.
   * SDPpack, a package of Matlab files currently in version 0.8 beta.
   * SDPSOL, a parser/solver for semidefinite programming and related
     determinant maximization problems with matrix structure.
   * SDPT3, a Matlab package.

See the semidefinite programming home pages maintained by Farid Alizadeh and
Christoph Helmberg for links and further information.

An extensive index of information on Global Optimization is maintained by
Arnold Neumaier of the Computational Mathematics group at the University of
Vienna. The following are a few of the codes available in this area:

   * BARON consists of a "core" module for global nonlinear optimization in
     continuous and/or discrete variables, and a variety of specialized
     modules for such problems as bilinear programming, fixed-charge
     programming, indefinite quadratic programming, linear multiplicative
     programming, and univariate polynomial programming. Information is
     available from Nick Sahinidis,

   * A software package for solving structured global optimization problems,
     cGOP, is available by ftp from the Computer-Aided Systems Laboratory at
     Princeton University. It implements a primal-dual decomposition
     algorithm applicable to general constrained biconvex problems, using a
     set of C subroutines to solve these problems via decomposition and
     branch-and-bound techniques; version 1.1 has been updated to use CPLEX
     4.0 and MINOS 5.4 as the solvers for various linear and convex
     subproblems. For assistance, write to

   * Fortran source code for global minimization using a stochastic
     integration algorithm (TOMS 667), is obtainable at

   * LGO integrates several global and local optimization solvers, which can
     be activated by the user in interactive or automatic execution modes.
     The PC version is embedded under a menu-driven interface.

The following products are said to do some nonlinear optimization, but don't
fall into any of the usual categories:

   * UniCalc, for "arbitrary algebraic systems of equations and
     inequalities," has been developed at the Russian Research Institute of
     Artificial Intelligence,

   * Fortran Calculus is a programming front end to a variety of nonlinear
     problem solvers, available from Optimal Designs,

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, containing a
forty-page introduction to the topic of GA. The Usenet newsgroup on GA,, has a FAQ on the topic, otherwise known as "The
Hitch-Hiker's Guide to Evolutionary Computation", available at Genetic Algorithm
code can be obtained at A commercial
organization, New Light Industries, Ltd. offers a "Genetic Algorithm Solver
for Excel" they call GENERATOR; their email address is
Simulated Annealing code for NLP and other problems is available at (or -- contact Lester Ingber
( for more info. A code called SDSC EBSA (Ensemble
Based Simulated Annealing) is available at, or contact Richard Frost
( 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.

Another technique that should be considered is "Constraint Programming"
(sometimes embedded in Prolog-like languages to form "Constraint Logic
Programming"). There is a Usenet newsgroup, comp.constraints, devoted to the
topic. A WWW page exists at Or you can
access the FAQ at // The
maintainer of that FAQ, Michael Jampel (, suggests CLP
is best suited for small problems that don't fit typical OR categories (LP,
QP, etc.),

   * "especially if there is indeterminism / incompleteness. Also, if you
     wish to mix numeric with non-numeric domains.... Also, if you need to
     do a lot of encoding of your problem to get it to fit into the OR
     technique; it may be better to use a relatively slow CSP technique on
     10 variables rather than a super-fast OR technique on 2^10 variables."

In the following table is a condensed version of a survey of NLP software
published in the April 1995 issue of " OR/MS Today", a publication of
INFORMS. For further information I would suggest you read the full article.
Several of the software vendors listed in the survey offer multiple
products, in keeping with the conventional wisdom that no one algorithm will
be best for all NLP models. Hence I have grouped the solver products by
vendor, rather than listing them alphabetically by product name. Since the
information won't fit on one line, I've broken the SOLVERS part of the table
into two pieces.

Key to Methods:
  SQP = Successive (Sequential) Quadratic Programming
  SLP = Successive (Sequential) Linear Programming
  GRG = Generalized Reduced Gradient

Solver Vendor                    Phone   E-mail address
Aptech Systems             206-432-7855
ARKI Consulting &
Development             +45 44-49-03-23
Frontline Systems          702-831-0300
ILOG                       415-390-9000
INRIA                    +33 13963-5557
Prof. L. Lasdon            512-471-9433
LINDO Systems              312-988-7422
Mathworks                  508-653-1415
NAG (Numerical
Algorithms Group)          630-971-2337
Rutherford Appleton
Laboratory              +44 1235-821900
SAITECH                    732-264-4700
Prof. K. Schittkowski   +49 921-55-3278
Stanford Business
Software                   415-962-8719
Prof. A.L. Tits            301-405-3669
Vanderplaats Research &
Development                415-962-8719
Visual Numerics            713-784-3131

Vendor             Product                 Method
Aptech             GAUSS CO module         SQP
ARKI               CONOPT                  GRG
Frontline Systems  GRG2, LSGRG             GRG
ILOG               Numerica                Constraint-based global search
INRIA              Scilab                  SQP
L. Lasdon          GRG2                    GRG
                   INTPT                   Interior point
                   SLP                     SLP
                   SQP                     SQP
LINDO Systems      LINGO                   GRG
Mathworks          NAG Foundation          various
                   Toolbox                 various
                   Optimization Toolbox
NAG                NAG Numerical Libraries various
Rutherford Lab     LANCELOT                Trust region, augmented
SAITECH            SOPT                    SQP, Interior point
K. Schittkowski    NLPQL                   SQP
                   others                  various
Stanford Bus Soft  LSSOL                   Active set method
                   MINOS                   Reduced gradient
                   NPSOL                   SQP
A.L. Tits          FSQP                    SQP
Vanderplaats       DOC/DOT                 various
Visual Numerics    IMSL                    various

Product          Phone  E-mail address
AIMMS    +31 23-5350935
AMPL       702-322-7600,
ASCEND     412-268-2344
CUTE      +32 81-724917
GAMS       415-583-8840

Here is a summary of other NLP codes mentioned in newsgroups in the past few
years (or, further information on the ones in the above table), sorted
alphabetically. Perhaps someone will volunteer to organize these references
more usefully.

   * Amoeba - Numerical Recipes
   * Brent's Method - Numerical Recipes
   * CO/CML - Applications written in GAUSS language, general constrained
     optimization and constrained maximum likelihood estimation using SQP.
     CML includes statistical inference (also Bayesian, bootstrap) for
     constrained parameters. Postscript file descriptions via Contact, or
     call +1-206-432-7855.
   * CONMAX - Fortran program for solving nonlinearly constrained problems
     of the form:
        o Choose x1,...,xn to minimize w subject to
        o abs(fi - gi(x1,...,xn)) .LE. w, 1 .LE. i .LE. m1
        o gi(x1,...,xn) .LE. w, m1 + 1 .LE. i .LE. m2
        o gi(x1,...,xn) .LE. 0, m2 + 1 .LE. i .LE. m3
     where the fi are given real numbers, and the gi are given smooth
     functions. Constraints of the form gi(x1,...,xn) = 0 can also be
     handled. Each iteration of our algorithm involves approximately solving
     a certain nonlinear system of first order ordinary differential
     equations to get a search direction. The program and its extensive
     user's guide can be obtained from the opt folder of netlib.
   * CONMIN - Vanderplaats, Miura & Associates, Colorado Springs, Colorado,
   * CONOPT - Large-scale GRG code, by Arne Drud. Available with GAMS,
     AIMMS, AMPL, LINGO, and What's Best (modeling languages - see LP FAQ)
     and as a standalone subroutine library. Phone +45 44 49 03 23, e-mail
   * DFPMIN - Numerical Recipes (Davidon-Fletcher-Powell)
   * Eureka - Borland Software (for IBM PC class of machines)
   * FFSQP/CFSQP (Fortran/C) - Contact Andre Tits, Free
     of charge to academic users. Solves 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). Interface to AMPL.
   * GENOCOP - Solves linearly constrained problems via a Genetic algorithm,
     available at Author: Zbigniew Michalewicz,
   * Harwell Library routines
        o VF01: based on R. Fletcher algorithm
        o VF02: based on M. Powell alogorithm
        o VF03: using "watchdog techniques" for line search. An improved
          version of VF02.
        o VF04: Automatically calculate 1st order derivatives, VF03 ia
          required to provide the derivatives.
   * Hooke and Jeeves algorithm - see reference below. A version is
     available at, and may be useful
     because it handles nondifferentiable and/or discontinuous functions.
   * LANCELOT - large-scale NLP. See the book by Conn et al. in the
     references section. For peaceful purposes only. For information on
     licensing this package, see the email addresses for Conn, Toint, or
     Gould, in the entry for CUTE,
   * 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.
   * MATLAB Optimization Toolbox - The Mathworks, Inc. 508-653-1415. Handles
     various nonlinear optimization problems. Data sheet available in
     postscript format at Email address: .
   * MINOS - Stanford Business Software Inc., 415-962-8719. Mailing address:
     2672 Bayshore Parkway, Suite 304, Mountain View, CA 94043. Email: This large-scale code is often used by
     researchers as a "benchmark" for others to compare with.
   * MINPACK I and II - Contact Jorge More',, or check Solves dense nonlinear least-squares
   * Nelder and Mead's method - Numerical Recipes, also Barabino.
   * NOVA - DOT Products, Houston TX
   * QLD - Contact: Solves Quadratic
     Programming and other nonlinear problems.
   * QPSOL - see LSSOL.
   * SLATEC - Quadratic solvers dbocls, dlsei, and other routines. National
     Energy Software Center, 9700 Cass Ave., Argonne, Illinois 60439. Also
     available at

An extremely useful book 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. A Web version is available, at least the parts that give info on
actual software packages, in URL

I would be 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:

   * 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)
   * 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).
   * 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.
   * 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.
   * 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.
   * 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.
   * P. Pardalos, "Generation of Large-Scale Quadratic Programs", ACM
     Transactions on Mathematical Software, Vol. 13, No. 2, p. 133.
   * P.M. Pardalos, Construction of test problems in quadratic bivalent
     programming, ACM Transactions on Mathematical Software, Vol. 17, No. 1
     (1991), pp. 74-87.
   * 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.
   * F. Schoen, "A Wide Class of Test Functions for Global Optimization", J.
     of Global Optimization, 3, 133-137, 1993, with C source code available
   * 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 or at 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:

   * Ingrid Bongartz
   * Andy Conn
   * Nick Gould
   * Philippe Toint

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

A collection of Global Optimization problems resides at In this directory,
(reverse.tar.Z) and (concave.tar.Z) contain a collection of test
problems for linear reverse convex programs, known as LRCP and concave
minimization problems. For further details, see the README file in the
directory, or contact Khosrow Moshirvaziri at

Fortran source code of global optimization test problems (1-10 dimensional)
are at the end of TOMS 667 fortran code, obtainable at

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 -]

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,
or use anonymous ftp at in
file genqp.code.tar.Z.

Hans D. Mittelmann and P. Spellucci have prepared a test environment of over
400 unconstrained and constrained nonlinear optimization problems, plus code
to facilitate interfacing solvers to them. This material is available as a
tar file from

[ ]

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, or are recent. I have *not* reviewed them

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
   * Harvey Greenberg has compiled an on-line Mathematical Programming

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

   * 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.

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