List of optimization software

Last updated October 07, 2024

Given a transformation between input and output values, described by a mathematical function, optimization deals with generating and selecting the best solution from some set of available alternatives, by systematically choosing input values from within an allowed set, computing the output of the function and recording the best output values found during the process. Many real-world problems can be modeled in this way. For example, the inputs could be design parameters for a motor, the output could be the power consumption. For another optimization, the inputs could be business choices and the output could be the profit obtained.

In continuous optimization, A is some subset of the Euclidean space Rⁿ, often specified by a set of constraints , equalities or inequalities that the members of A have to satisfy. In combinatorial optimization, A is some subset of a discrete space, like binary strings, permutations, or sets of integers.

The use of optimization software requires that the function f is defined in a suitable programming language and connected at compilation or run time to the optimization software. The optimization software will deliver input values in A, the software module realizing f will deliver the computed value f(x) and, in some cases, additional information about the function like derivatives.

In this manner, a clear separation of concerns is obtained: different optimization software modules can be easily tested on the same function f, or a given optimization software can be used for different functions f.

The following tables provide a list of notable optimization software organized according to license and business model type.

Free and open-source software

Applications

Name	License	Description
ADMB	BSD	nonlinear optimization framework using automatic differentiation.
ASCEND	GPL	mathematical modelling chemical process modelling system.
CUTEr	GPL	testing environment for optimization and linear algebra solvers.
GNU Octave	GPL	software package using a high-level programming language, primarily intended for numerical computations; it is mostly compatible with MATLAB.
Scilab	CeCILL	cross-platform numerical computational package and a high-level, numerically oriented programming language with a numerical optimization framework.

Software libraries

Name	License	Description
ALGLIB	GPL	dual licensed (GPL/commercial) optimization library (LP, QP and nonlinear programming problems), optionally using automatic differentiation. Cross-language: C++, C#.
COIN-OR	EPL 1.0	integer programming, linear programming, nonlinear programming.
Dlib	BSL‑1.0	unconstrained/box-constrained nonlinear/QP optimization library written in C++.
GEKKO	MIT	machine learning and optimization of mixed-integer and differential algebraic equations in Python.
GLPK	GPL	GNU Linear Programming Kit with C API.
HiGHS	MIT	linear programming (LP), mixed integer programming (MIP), and convex quadratic programming (QP).^[1]
IPOPT	EPL (was CPL)	large scale nonlinear optimizer for continuous systems (requires gradient), C++ (formerly Fortran and C). It became a part of COIN-OR.^[2]
MINUIT (now MINUIT2)	LGPL	unconstrained optimizer internally developed at CERN.
OpenMDAO	Apache License	Multidisciplinary Design, Analysis, and Optimization (MDAO) framework, written in Python. The development is led out of the NASA Glenn Research Center, with support from the NASA Langley Research Center.
SCIP	Apache License	solver for mixed integer programming (MIP) and mixed integer nonlinear programming (MINLP).
SciPy	BSD	general numeric package for Python, with some support for optimization.
Uno	MIT	Lagrange-Newton solver that unifies nonconvex optimization, implemented in C++. Developed at Argonne National Laboratory and Zuse Institute Berlin. ^[3]

Proprietary software

AIMMS – optimization modelling system, including GUI building facilities.
ALGLIB – dual licensed (GPL/commercial) constrained quadratic and nonlinear optimization library with C++ and C# interfaces.
Altair HyperStudy – design of experiments and multidisciplinary design optimization.
AMPL – modelling language for large-scale linear, mixed integer and nonlinear optimization.
ANTIGONE – a deterministic global optimization MINLP solver.
APMonitor – modelling language and optimization suite for large-scale, nonlinear, mixed integer, differential, and algebraic equations with interfaces to MATLAB, Python, and Julia.
Artelys Knitro – large scale nonlinear optimization for continuous and mixed-integer programming.
ASTOS – AeroSpace Trajectory optimization Software for launch, re-entry, and generic aerospace problems.
BARON – optimization of algebraic nonlinear and mixed-integer nonlinear problems.
COMSOL Multiphysics – a cross-platform finite element analysis, solver and multiphysics simulation software.
CPLEX – solver for linear and quadratic programming with continuous or integer variables (MIP).
FEATool Multiphysics – FEA GUI Toolbox for MATLAB.
FICO Xpress – solver for linear and quadratic programming with continuous or integer variables (MIP).
FortMP – linear and quadratic programming.
FortSP – stochastic programming.
GAMS – General Algebraic Modeling System.
Gurobi Optimizer – solver for linear and quadratic programming with continuous or integer variables (MIP).
HEEDS MDO – multidisciplinary design optimization using SHERPA, a hybrid, adaptive optimization algorithm.
IMSL Numerical Libraries – linear, quadratic, nonlinear, and sparse QP and LP optimization algorithms implemented in standard programming languages C, Java, C# .NET, Fortran, and Python.
IOSO – (Indirect optimization on the basis of Self-Organization) a multi-objective, multidimensional nonlinear optimization technology.
Kimeme – an open platform for multi-objective optimization and multidisciplinary design optimization.
LINDO – (Linear, Interactive, and Discrete optimizer) a software package for linear programming, integer programming, nonlinear programming, stochastic programming, and global optimization. The "What's Best!" Excel add-in performs linear, integer, and nonlinear optimization using LINDO.
LIONsolver – an integrated software for data mining, analytics, modelling Learning and Intelligent OptimizatioN and reactive business intelligence approach.
modeFRONTIER – an integration platform for multi-objective and multidisciplinary optimization, which provides a seamless coupling with third party engineering tools, enables the automation of the design simulation process, and facilitates analytic decision-making.
Maple – linear, quadratic, and nonlinear, continuous and integer optimization. Constrained and unconstrained. Global optimization with add-on toolbox.
MATLAB – linear, integer, quadratic, and nonlinear problems with Optimization Toolbox; multiple maxima, multiple minima, and non-smooth optimization problems; estimation and optimization of model parameters.
MIDACO a lightweight software tool for single- and multi-objective optimization based on evolutionary computing. Written in C/C++ and Fortran with gateways to Excel, VBA, Java, Python, Matlab, Octave, R, C#, and Julia.
Mathematica – large-scale multivariate constrained and unconstrained, linear, quadratic and nonlinear, continuous, and integer optimization.
ModelCenter – a graphical environment for integration, automation, and design optimization.
MOSEK – linear, quadratic, conic and convex nonlinear, continuous, and integer optimization.
NAG – linear, quadratic, nonlinear, sums of squares of linear or nonlinear functions; linear, sparse linear, nonlinear, bounded or no constraints; local and global optimizations; continuous or integer problems.
NMath – linear, quadratic and nonlinear programming.
Octeract Engine – a deterministic global optimization MINLP solver. Plans exist for additional features.
OptimJ – Java-based modelling language. Premium Edition includes support for Mosek and CPLEX solvers.
Optimus platform – a process integration and design optimization platform developed by Noesis Solutions.
optiSLang – software solutions for CAE-based sensitivity analysis, optimization, and robustness evaluation.
OptiY – a design environment providing modern optimization strategies and state of the art probabilistic algorithms for uncertainty, reliability, robustness, sensitivity analysis, data-mining, and meta modelling.
OptiStruct – award-winning CAE technology for conceptual design synthesis and structural optimization.
OptQuest – metaheuristics-based optimization plugin for simulation-based optimization in conjunction with discrete-event simulation software.
PottersWheel – parameter estimation in ordinary differential equations (MATLAB toolbox, free for academic use).
pSeven – software platform for automation of engineering simulation and analysis, multidisciplinary optimization and data mining, developed by DATADVANCE.
SAS – a software suite developed by SAS Institute for advanced analytics (statistics, forecasting, machine learning, optimization, etc.), business intelligence, customer intelligence, data management, risk management, and many more.
SmartDO – multidisciplinary global design optimization, specialized in computer-aided engineering (CAE). using the direct global search approaches.
SNOPT – large-scale optimization problems.
The Unscrambler – product formulation and process optimization software.
TOMLAB – supports global optimization, integer programming, all types of least squares, linear, quadratic, and unconstrained programming for MATLAB. TOMLAB supports solvers like CPLEX, SNOPT, KNITRO and MIDACO.
VisSim – a visual block diagram language for simulation and optimization of dynamical systems.
WORHP – a large-scale sparse solver for continuous nonlinear optimization.

Freeware/free for academic use

AIMMS
AMPL
APMonitor – free for academic and commercial use alike, with Python and MATLAB integrations.
ASTOS
CPLEX
Couenne – An open source solver for the deterministic global optimization of MINLPs licensed under the Eclipse Public License.
FICO Xpress
Galahad library
Gekko
Gurobi Optimizer - free for academic users
LIONsolver
MIDACO – a software package for numerical optimization based on evolutionary computing.
MINTO – integer programming solver using branch and bound algorithm; freeware for personal use.
MOSEK – a large scale optimization software. Solves linear, quadratic, conic and convex nonlinear, continuous and integer optimization.
OptimJ – Java-based modelling language; the free edition includes support for lp_solve, GLPK and LP or MPS file formats.
PottersWheel – parameter estimation in ordinary differential equations (free MATLAB toolbox for academic use).
Pyomo – collection of Python software packages for formulating optimization models.
UFO Fortran package for numerical optimization
WORHP

Related Research Articles

Mathematical optimization or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.

In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints are not linear equalities or the objective function is not a linear function. An optimization problem is one of calculation of the extrema of an objective function over a set of unknown real variables and conditional to the satisfaction of a system of equalities and inequalities, collectively termed constraints. It is the sub-field of mathematical optimization that deals with problems that are not linear.

<span class="mw-page-title-main">AMPL</span> Algebraic modeling language

AMPL is an algebraic modeling language to describe and solve high-complexity problems for large-scale mathematical computing . It was developed by Robert Fourer, David Gay, and Brian Kernighan at Bell Laboratories. AMPL supports dozens of solvers, both open source and commercial software, including CBC, CPLEX, FortMP, MOSEK, MINOS, IPOPT, SNOPT, KNITRO, and LGO. Problems are passed to solvers as nl files. AMPL is used by more than 100 corporate clients, and by government agencies and academic institutions.

The general algebraic modeling system (GAMS) is a high-level modeling system for mathematical optimization. GAMS is designed for modeling and solving linear, nonlinear, and mixed-integer optimization problems. The system is tailored for complex, large-scale modeling applications and allows the user to build large maintainable models that can be adapted to new situations. The system is available for use on various computer platforms. Models are portable from one platform to another.

SNOPT, for Sparse Nonlinear OPTimizer, is a software package for solving large-scale nonlinear optimization problems written by Philip Gill, Walter Murray and Michael Saunders. SNOPT is mainly written in Fortran, but interfaces to C, C++, Python and MATLAB are available.

The TOMLAB Optimization Environment is a modeling platform for solving applied optimization problems in MATLAB.

The Zuse Institute Berlin is a research institute for applied mathematics and computer science on the campus of Freie Universität Berlin in Dahlem, Berlin, Germany.

MOSEK is a software package for the solution of linear, mixed-integer linear, quadratic, mixed-integer quadratic, quadratically constrained, conic and convex nonlinear mathematical optimization problems. The applicability of the solver varies widely and is commonly used for solving problems in areas such as engineering, finance and computer science.

APOPT is a software package for solving large-scale optimization problems of any of these forms:

Optimization Toolbox is an optimization software package developed by MathWorks. It is an add-on product to MATLAB, and provides a library of solvers that can be used from the MATLAB environment. The toolbox was first released for MATLAB in 1990.

LINDO is a software package for linear programming, integer programming, nonlinear programming, stochastic programming and global optimization.

<span class="mw-page-title-main">OptiSLang</span>

optiSLang is a software platform for CAE-based sensitivity analysis, multi-disciplinary optimization (MDO) and robustness evaluation. It was originally developed by Dynardo GmbH and provides a framework for numerical Robust Design Optimization (RDO) and stochastic analysis by identifying variables which contribute most to a predefined optimization goal. This includes also the evaluation of robustness, i.e. the sensitivity towards scatter of design variables or random fluctuations of parameters. In 2019, Dynardo GmbH was acquired by Ansys.

Deterministic global optimization is a branch of mathematical optimization which focuses on finding the global solutions of an optimization problem whilst providing theoretical guarantees that the reported solution is indeed the global one, within some predefined tolerance. The term "deterministic global optimization" typically refers to complete or rigorous optimization methods. Rigorous methods converge to the global optimum in finite time. Deterministic global optimization methods are typically used when locating the global solution is a necessity, when it is extremely difficult to find a feasible solution, or simply when the user desires to locate the best possible solution to a problem.

Artelys Knitro is a commercial software package for solving large scale nonlinear mathematical optimization problems.

ANTIGONE, is a deterministic global optimization solver for general Mixed-Integer Nonlinear Programs (MINLP).

Octeract Engine is a proprietary massively parallel deterministic global optimization solver for general Mixed-Integer Nonlinear Programs (MINLP).

References

↑ Hall, Julian (21 September 2020). HiGHS: High-performance open-source software for linear optimization (PDF). Edinburgh, United Kingdom: University of Edinburgh. Retrieved 2022-02-27. Presentation.
↑ "Projects". COIN-OR: Computational Infrastructure for Operations Research. 8 October 2014. Retrieved 10 March 2021.
↑ Vanaret, Charlie; Leyffer, Sven (5 September 2024). Uno, a next-gen solver for unifying nonlinearly constrained nonconvex optimization. Argonne National Laboratory and Zuse Institute Berlin. Presentation.

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[hall-2020-1] Hall, Julian (21 September 2020). HiGHS: High-performance open-source software for linear optimization (PDF). Edinburgh, United Kingdom: University of Edinburgh. Retrieved 2022-02-27. Presentation.

[coin-or-projects-2] "Projects". COIN-OR: Computational Infrastructure for Operations Research. 8 October 2014. Retrieved 10 March 2021.

[vanaret-2024-3] Vanaret, Charlie; Leyffer, Sven (5 September 2024). Uno, a next-gen solver for unifying nonlinearly constrained nonconvex optimization. Argonne National Laboratory and Zuse Institute Berlin. Presentation.

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