Global Optimization Software

Global Search, As Timely As Ever

``Consider everything. Keep the good. Avoid evil whenever you notice it.''
(1 Thess. 5:21-22)

This file is part of my global optimization web site. It contains sections on

Branch and Bound Codes
Solvers based on algebraic techniques
Other Deterministic Codes
Simulated Annealing Codes
Evolutionary Algorithm Codes
Other Stochastic Codes
Specialized Codes

Some Commercial Systems

We are working on a comparative evaluation of a number of currently available constrained global optimization programs. The results will be made publicly available here. If, as an authors of such software, you are interested in our evaluation of your code, the notes on Global Optimization Software Evaluation will tell you what you need to know.

Janos Pintér's recent Optima Newsletter survey on continuous global optimization software is now available online. Please send new or updated information regarding your global optimization software to Janos Pintér.

Below is a list of publicly available global optimization programs (and a few commercial ones). The quality in terms of speed, reliability, and/or automatic parameter choices, stopping tests and ease of use differs widely. I have not checked out the codes (will perhaps be done some day); but I am very interested in comparisons (cf. test results available online) or any other useful comments (mail here) on the sources given.

In a few cases, I added information on problems where a code has difficulties. This is not meant to criticize the code (it may work excellent on other classes of problems) but as a service to the research community: to challenge researchers to overcome current limitations, and to compete for better and better techniques.

Lack of critical information doesn't mean that a code has no problems, but only that nobody informed me about any weaknesses. Indeed, all current global optimization codes for nonlinear problems are unreliable (i.e., cannot always find the global optimum in a reasonable, predictable amount of time and space), with unknown performance limits.

At the present stage of knowledge, if you need global optimization you must experiment with the known methods (or packages) until you have something satisfying you. Unfortunately, this is a rather unsatisfactory state of the art... In general, performance must be expected to deteriorate with increasing dimension. Dimension 50-100 is probably already very high for general-purpose codes on all but the simplest problems. Many codes allow no constraints or only bound constraints.

Since I was asked repeatedly what to try first, here are some suggestions for continuous problems. But they are backed up by little else than my feelings of what I'd try first, and I am using a small font size to emphasize that this is information of a poorer quality than in the rest of my pages. Please write me if you have significant experience that leads to improved suggestions, or that confirms the ones given.

If the problem has the particular structure required for cGOP, BARON, or Q01SUBS, try these.
Otherwise, try Boender-Timmer-Rinnoy-Kan (for unconstrained problems), ASA (for bound constrained problems), or GENOCOP III (for general constrained problems).
In a comparison of several stochastic algorithms in Fortran or C on 45 problems of dimension 2-30, Tibor Csendes' implementation of the Boender-Timmer-Rinnoy-Kan algorithm was the clear winner. (But MCS and DIRECT/gblSolve that are publicly available in Matlab only and did not fit the environment of the above study are even better, at least on the limited Dixon-Szego test set.) So you might want to use a penalty or barrier formulation of a constrained problem to be able to apply one of these algorithms.
If these first choices turn out to be not reliable enough on your problem class, but the problem is in factorable form (i.e., uses arithmetic operations and elementary functions but no branches), try INTOPT_90. These are generally slow since they use directed rounding; but, unless the program is overwhelmed by the number of boxes to be processed, you can be sure with mathematical certainty that you found all global minima!

Public Domain Systems

A list of global optimization codes in the public domain (or at least free for academic users). Please respect the author's conditions for using and/or modifying the codes.

Branch and Bound Codes

(These codes are often combined with range reduction techniques using interval analysis or constraint satisfaction)

interalg, interval global solver for nonlinear programming (in Python, by Dmitrey Kroshko)

BMIBNB, Global solver for nonconvex problems in the YALMIP modeling environment (in Matlab, by Johan Löfberg)

ICOS, rigorous complete solver for continuous constraint satisfaction problems

LaGO, Branch and Cut algorithm for nonconvex mixed integer nonlinear programs

RSOLVER (by Stefan Ratschan)
for solving quantified inequality constraints. Examples include: projecting the solution set of a set of inequality constraints to two dimensions.

GANSO, programming library for global and non-smooth optimization with linear constraints (with both complete and probabilistic methods, free for up to 20 variables and up to 20 constraints)

Concave, concave programming with linear constraints (by Alexander Rusakov)

LP_SOLVE, Sparse Mixed Integer Linear Programming (by Michel Berkelaar), a
Matlab Interface (by Kantor and Chen; get lpmex.tar and lpmex_ext.tar), and an
Online Interface (by Dmitry Golovashkin)

LINDO, LINGO for linear and nonlinear mixed integer programs (small-scale versions available for free)

Visual XPRESS Linear mixed integer programming (free student edition, for models with at most 100 rows and 200 variables, and tables and variables having at most two dimensions)

MODLER integer linear programming in DOS (by Harvey Greenberg)

ABACUS, A Branch-And-Cut System for combinatorial and linear mixed integer optimization (by Stefan Thienel)
(calls the commercial Cplex or SoPlex packages)

Mixed Integer LP-optimizer (Borland Pascal, by Markus Weidenauer)

setconst, NLP solver with set constrained variables in Matlab (assumes that a solver for the NLP with set constraints replaced by bound constraints is available) NLBranch, Mixed Integer Nonlinear Programming Solver
(seems to assume that for fixed discrete parameters, the problem is convex)

cGOP, Global Optimization Program in C (by Visweswaran and Floudas)
Solves global optimization problems with an objective function of the form
a^Tx+b^Ty+x^TAy+f_1(x)+f_2(y) with convex f_1, f_2,
and linear constraints. [Any linearly constrained problem with objective function of the form `quadratic+convex' can be brought into this form by introducing additional variables.]
Requires the commercial codes MINOS 5.4 and (or?) CPLEX 3.0 to solve subproblems.
``cGOP has been in use at the Computer-Aided Systems Laboratory in Princeton University for the last two years, and has been used to solve problems involving several hundred variables and constraints. Currently, versions are available for Hewlett-Packard, Silicon Graphics and IBM RS6000 machine architectures.''

BOB: Branch-and-bound Optimization liBrary (for combinatorial optimization)

INTOPT_90 Interval Branch and Bound ands its sequel GlobSol in Fortran 90 (by Baker Kearfott)
for global optimization with general factorable constraints, with rigorously guaranteed results (even roundoff is accounted for correctly)
Parallel version (with MPI, by Chenyi Hu)

Reduced Space Branch and Bound in C++ (by Tom Epperly)
for global optimization with general factorable constraints [link dead]

PROFIL/BIAS Interval Branch and Bound (by Jansson and Knüppel)
bound constrained global optimization, with rigorously guaranteed results (even roundoff is accounted for correctly)
C-code for PC and a number of UNIX platforms, because of machine dependence in the interval arithmetic
The code has problems for certain rational objective functions (such as Wood, Kowalik-Osborne, Bard from the Moré/Garbow/Hillstrom test set), where the natural interval extension results in excessive overestimation

Concave programming (by Alexander Rusakov)

Q01SUBS (by Panos Pardalos)
minimizes sparse quadratic functions subject to {0,1}-constraints. Concave quadratic problems with finite two-sided bound constraints can be rephrased in this way.

Portable Parallel Branch-and-Bound Library (by Stefan Tschoeke)

BOB, Parallel Branch-and-bound Optimization Library

RSOLVER (by Stefan Ratschan)
for solving quantified inequality constraints. Examples include: projecting the solution set of a set of inequality constraints to two dimensions.

QEPCAD, Quantifier Elimination by Partial Cylindrical Algebraic Decomposition (mathematically rigorous)

INTSOLVER: An interval based solver for Global Optimization (in Matlab, by T. Montanher)

INTBIS (by Baker Kearfott)
finds all solutions of polynomial systems of equations, with rigorously guaranteed results (even roundoff is accounted for correctly)

IAsolver in Java (Brandeis Interval Arithmetic Constraint Solver)

zerOne vertex enumeration code for 0/1 polytopes

OPBDP, implicit enumeration for 0-1 problems in C++ (by Peter Barth)
Logic-Based 0-1 Constraint Programming

clp(FD), Constraint Logic Programming over Finite Domains

CUTSDP Toolbox for a Cutting-Plane Approach Based on Semidefinite Programming, with three applications: max-cut, graph bisection, and graph equipartition (by Stefan Karisch)

VBCTOOL Visualization of Branch and Cut algorithms

Solvers based on algebraic techniques

SparsePOP

GloptiPoly (in Matlab, by Henrion and Lasserre)
globally ''minimizing a multivariable polynomial function subject to polynomial inequality, equality or integer constraints.''
A related recent Matlab 6 program is SOSTOOLS for sums of squares optimization problems.

Global Minima of Polynomial Optimization Problems using Gröbner Bases (by Kristoffer Hägglöf)

CMPSm, Continuation Method for Polynomial Systems (in Matlab)
for finding all isolated solutions of a system of polynomial equations

Other Deterministic Codes

GLS, Global Line Search

Black Box Optimization with Data Analysis for the global optimization of smooth problems with expensive objective and/or constraints (by Kevin Kofler)

MCS, Multilevel Coordinate Search a Matlab program for bound constrained global optimization using function values only (by Huyer and Neumaier)

NLopt algorithms for derivative-free global optimization

Jörg Gablonsky's Fortran implementation of DIRECT, Divide Rectangles, a simple global optimization method for bound constrained problems by Jones, Perttunen and Stuckmann

Direct.m, a Matlab implementation of DIRECT

14 Binary Constraint Satisfaction Programs in C (by Peter van Beek)
The recommended method is FC_CBJ (= number 11).

ECLiPSe, a development environment for constraint programming applications

TOMS/647 Quasirandom Sequence Generator (29K, in Fortran 77) with low discrepancy for deterministic global search in dimensions 1-40

TOMS/738 Niederreiter's Low Discrepancy Sequences (127K, in Fortran 77) for deterministic global search in arbitrary dimensions

Complementarity Toolbox for MATLAB (by Mike Ferris)

Inductive Search Algorithm in C++ (by George Bilchev)
Very fast, but may fail occasionally. Got the first place at the 1st International Contest on Evolutionary Computation on a real-valued function testsuite.

Solution of integer systems of linear equations in Fortran (by J. Springer)

0-1 Multiple Knapsack Problem in Fortran (by Martello and Toth)

CGU, Convex Global Underestimation Code for protein folding (by Andy Phillips)

Global Optimization Using Hyperbolic Cross Points (by Novak and Ritter)

Simulated Annealing Codes

CSA, constrained simulated annealing with AMPL interface

SIMMAN, Simulated Annealing Code in Fortran (37K, by Bill Goffe)
from netlib; handles bound constraints

ASA, CalTech Adaptive Simulated Annealing Code in C (by Lester Ingber) and a Matlab Interface
handles bound constraints

EBSA, Ensemble Based Simulated Annealing in C (by Richard Frost)

SA, Simulated Annealing Code in C++, C, Ada and Forth (by Everett Carter)

Evolutionary Algorithm Codes

Global Optimization Algorithms - Theory and Application

List of References on Constraint-Handling Techniques used with Evolutionary Algorithms

A Commented List of Genetic Algorithm Codes

another one

an uncommented one

Differential Search Algorithm (DSA) for unconstrained optimization (by Pinar Civicioglu)

PSwarm, global optimization for bound constrained and linearly constrained problems
with interfaces to AMPL, Python, R

CMA-ES, Covariance Matrix Adaptation Evolution Strategy for bound constrained optimization (by Nikolaus Hansen)
Best code in the 2005 IEEE Congress on Evolutionary Computation Benchmark (with code for the comparison)

Constrained differential evolution for constained nonlinear programming (by Tetsuyuki Takahama)
Best code in the 2006 IEEE Congress on Evolutionary Computation Benchmark

GENOCOP III, Genetic Algorithm for Constrained Problems in C (by Zbigniew Michalewicz)
GENOCOP III Code (60K)
handles general nonlinear constraints using quadratic penalty terms

DE, Differential Evolution Genetic Algorithm (local copy, with an interface for sequential use)
for unconstrained problems, in Matlab and C (by Rainer Storn)
(The bounds to be provides only delineate the initial search region.)
DE has won the third place at the 1996 1st International Contest on Evolutionary Computation on a real-valued function testsuite. DE was the best genetic algorithm approach. (The first two places of the contest were won by non-GA algorithms.)

Matlab Code for Biomimicry for Optimization

PIKAIA, genetic algorithm in Fortran77/90 (by Charbonneau,Knapp and Miller)

PGAPack, Parallel Genetic Algorithm in Fortran and C, with an extensive test set (from Argonne National Laboratory)

CGA, A C Genetic Algorithm

GAS, Genetic Algorithm in C++ (by Jelasity and Dombi)

GAOT, Genetic Algorithms Optimization Toolbox in Matlab (by Jeffrey A. Joines)

Genetic Algorithm in Matlab (by Michael B. Gordy)

GAucsd, Genetic Algorithm in C (from University of California, San Diego)

Genesys Genetic Algorithm in C (tar.Z, 708K, by John Grefenstette)

GAlib, C++ Genetic Algorithm Library (by Matthew Wall)

GAC Genetic Algorithm in C and Common Lisp (by Bill Spears)

GAGA Genetic Algorithm for General Application in C (by Ian Poole)

PARAGenesis Parallel Genetic Algorithm in C* (tar.Z, 121K, by Michael van Lent)

Evolutionary Algorithms for MATLAB (links by Hartmut Pohlheim)

Particle swarm and simulated annealing codes (by Brecht Donckels) [download links currently unaccessible]

Other Stochastic Codes

Derivative-Free Global Optimization

BIPOP-CMAES, one of the best stochastic global solvers in 2009 (if enough function values are permitted)

space, Stochastic Process Analysis of Computer Experiments (bound constrained optimization of very expensive functions of few variables, in C++, by Matthias Schonlau)

Stochastic ranking for constrained evolutionary optimization (Matlab code by Runarsson and Yao)

GLOBAL, Derivative-Free Boender-Timmer-Rinnoy Kan Algorithm in Fortran 77 and C (by Tibor Csendes)
Fortran 95 version (translated by Alan Miller)

StoGO, stochastic bound constrained global optimization using gradients (in C++)

The SGOPT Optimization Library (by William Hart)
``The SGOPT optimization library provides an object-oriented interface to a variety of optimization algorithms, especially stochastic optimization methods used for global optimization.... SGOPT includes a generic C++ genetic algorithm facility that is similar to the GAlib package. Additionally, it contains mechanisms for GA-local search hybrids, as well as an implementation of evolutionary pattern search algorithms.''

StoGO in C++ , branch and probability bound code for bound constrained problems, needs function values and gradients

IHR, Improving-Hit & Run Algorithm in Fortran (by Zelda Zabinsky)
allows integer constraints and general inequality constraints (assuming a strictly feasible point exists)

SPSA, Simultaneous Perturbation Stochastic Approximation (Matlab code for noisy global optimization, by James Spall)

TOMS/744, Stochastic algorithm for global minimization with constraints in Lisp (by F. M. Rabinowitz)

TOMS/667 in Fortran77 (210K, by Aluffi-Pentini, Parisi and Zirilli)
global optimization via stochastic integration;
with Fortran code of low-dimensional global optimization test problems

Bayesian Global Optimization in Fortran and C++ (by Audris Mockus)

Level Set Programming for Global Optimization (zip, 79K)
( Austrian Mirror Site)

Integer Local Search: WSAT(OIP)

LIONsolver, reactive search for

Other Heuristic Codes

Reactive Tabu Search

Specialized Codes

Concorde

GRASP, Dense Quadratic Assignment Problems

GRASP, Quadratic Assignment and Set Covering Heuristics

QAPP, Quadratic Assignment Branch and Bound

Facility Layout and Location MATLAB Toolbox (by Michael G. Kay)
includes a mixed integer LP solver (based on the Matlab optimization toolbox)

Some Commercial Systems

(no attempt for completeness is made) GloMIQO, Global Mixed-Integer Quadratic Optimizer in GAMS (by Misener and Floudas)
solves quadratically constrained quadratic programs with optional integer constraints to global optimality.

BARON, Branch-And-Reduce Optimization Navigator in Fortran (by Nikos Sahinidis)
A general purpose solver for optimization problems with nonlinear constraints and/or integer variables. Fast specialized solvers for many linearly constrained problems. GAMS interface.

LINDO/LINGO now has global optimization facilities

Interval Global Solver from Frontline Systems. Visual Basic and Excel interfaces

LSGRG Nonlinear Mixed Integer Solver for AMPL

MINOPT (Mixed Integer Nonlinear Programming)

GAMS/DICOPT Solver for Mixed Integer Nonlinear Programming

EZMod Quadratic Mixed Integer Programming (QMIP) Solver

CPLEX Linear Mixed Integer Solver (branch-and-bound for mixed integer linear programs)

CMMIP, Mixed Integer Programming for mixed integer LPs
``In CMMIP, each processor runs a version of CPLEX ... Parallelism is attained by looking at multiple nodes of the branch-and-bound search tree simultaneously ... near-linear speedups on up to 128 processors''

MINTO, Mixed INTeger Optimizer for mixed integer LPs

IBM Optimization Subroutine Library (OSL) (with a branch-and-bound mixed integer QP solver)

NAG FortMP (branch-and-bound mixed integer LP solver)

XPRESS-MP (mixed integer LP, QP and MINLP solver)

TOMLAB modeling and optimization environment. It includes, apart from many local algorithms, improved versions of the global optimization algorithms DIRECT and EGO of Jones, rbfSolve of Gutmann and Powell, Pinter's LGO, and mixed integer nonlinear solvers of Leyffer and Fletcher.

OptQuest (Tabu and Scatter search, etc.)

OptiREX, Cutting optimizer for rectangular sheets

ILOG Solver C++ software library which solves industrial problems for which generating or optimizing a solution is a highly combinatorial task, using constraint satisfaction techniques

Koalog Constraint Solver in Java

HEURO, Integrated Optimization Environment (for large scale black box global optimization with expensive objective function)

Constrained local and global optimization (by Janos Pinter)
includes MathOptimizer for Mathematica and LGO on various platforms

IOSO, Indirect Optimization Based on Self-Organization in Fortran 77, for large-scale and possibly multiobjective inequality constrained problems
heuristic: builds local statistical models from function values and optimizes these

Constrained Global Optimization with Mathematica (including mixed integer problems and 0-1 integer problems) using adaptive grid refinement (by Loehle Enterprises, commercial)

Generator, Genetic Algorithm (for Excel worksheets)
``You can use Generator to maximize profits...'' (... of New Light Industries, Ltd., of course)

Magestic, Data Fitting by Global Optimization (for Excel worksheets, commercial)

GLO Global Local Optimizer (by Michael J. Murphy)
genetic algorithm for expensive black box (finite element) objective functions with constraints

Pointer, Constrained Optimization Tool with automatic selection or combination of up to four different search algorithms, including their tuning parameters (by Synaps, Inc.)

OptiA
``Options for global optimization include algorithms based on cluster analysis and Monte Carlo techniques.''

Curvi for bound constrained optimization (by Windward Technologies)

OPTECH guided stochastic search for constrained global optimization

GEATbx Matlab Genetic and Evolutionary Algorithm Toolbox (by Hartmut Pohlheim)

Astrokettle Bin Packing in 2D and 3D

HIRON, stochastic global search for multivariate functions on a box

GIRAFFE Production Scheduling

UNICALC Interval Branch and Bound (by Alexander Semenov)
accepts inequality and/or equality constraints and decision variables (free demo version)

CLP(BNR), Constraint Logic Programming for Boolean, Natural and Real variables

ECLiPSe, ECRC Constraint Logic Parallel System constraint satisfaction (cheap for academic users)

IF/Prolog Prolog constraint satisfaction package

CHIP Prolog, C, and C++ programs for constraint satisfaction and scheduling