Given a transformation between input and output values, described by a

mathematical function In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the func ...

''f'',

optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...

deals with generating and selecting a 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 can be design parameters of a motor, the output can be the power consumption, or the inputs can be business choices and the output can be the obtained profit. An

optimization problem In mathematics, computer science and economics, an optimization problem is the problem of finding the ''best'' solution from all feasible solutions. Optimization problems can be divided into two categories, depending on whether the variables ...

, in this case a minimization problem, can be represented in the following way :''Given:'' a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

''f'' : ''A''

\to

R from some set ''A'' to the

real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

s :''Search for:'' an element ''x''₀ in ''A'' such that ''f''(''x''₀) ≤ ''f''(''x'') for all ''x'' in ''A''. In continuous optimization, ''A'' is some

subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...

of the

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...

R^''n'', often specified by a set of ''

constraint Constraint may refer to: * Constraint (computer-aided design), a demarcation of geometrical characteristics between two or more entities or solid modeling bodies * Constraint (mathematics), a condition of an optimization problem that the solution ...

s'', equalities or inequalities that the members of ''A'' have to satisfy. In combinatorial optimization, ''A'' is some

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 compile 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

Software libraries

Proprietary software

AIMMS AIMMS (acronym for Advanced Interactive Multidimensional Modeling System) is a prescriptive analytics software company with offices in the Netherlands, United States, China and Singapore. It has two main product offerings that provide modeling and ...

– optimization modeling system, including GUI building facilities. *

ALGLIB ALGLIB is a cross-platform open source numerical analysis and data processing library. It can be used from several programming languages ( C++, C#, VB.NET, Python, Delphi). ALGLIB started in 1999 and has a long history of steady development wit ...

– dual licensed (GPL/commercial) constrained quadratic and nonlinear optimization library with C++ and C# interfaces. * Altair HyperStudy – design of experiments and multi-disciplinary design optimization. *

AMPL AMPL (A Mathematical Programming Language) is an algebraic modeling language to describe and solve high-complexity problems for large-scale mathematical computing (i.e., large-scale optimization and scheduling-type problems). It was developed b ...

– modelling language for large-scale linear, mixed integer and nonlinear optimization. *

ANTIGONE In Greek mythology, Antigone ( ; Ancient Greek: Ἀντιγόνη) is the daughter of Oedipus and either his mother Jocasta or, in another variation of the myth, Euryganeia. She is a sister of Polynices, Eteocles, and Ismene.Roman, L., & Roma ...

– a deterministic global optimization MINLP solver. *

APMonitor Advanced process monitor (APMonitor) is a modeling language for differential algebraic ( DAE) equations. It is a free web-service or local server for solving representations of physical systems in the form of implicit DAE models. APMonitor is ...

– modeling language and optimization suite for large-scale, nonlinear, mixed integer, differential and algebraic equations with interfaces to MATLAB, Python, and Julia. *

Artelys Knitro Artelys Knitro is a commercial software package for solving large scale nonlinear mathematical optimization problems. KNITRO – (the original solver name) short for "Nonlinear Interior point Trust Region Optimization" (the "K" is silent) – wa ...

– large scale nonlinear optimization for continuous and mixed-integer programming. * ASTOS – AeroSpace Trajectory Optimization Software for launcher, re-entry and generic aerospace problems. * BARON – optimization of algebraic nonlinear and mixed-integer nonlinear problems. * COMSOL Multiphysics – a cross-platform

finite element The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical models, mathematical modeling. Typical problem areas of interest include the traditional fields of struct ...

analysis, solver and

multiphysics In computational modelling, multiphysics simulation (often shortened to simply "multiphysics") is defined as the simultaneous simulation of different aspects of a physical system or systems and the interactions among them. For example, simultaneous ...

simulation software Simulation software is based on the process of modeling a real phenomenon with a set of mathematical formulas. It is, essentially, a program that allows the user to observe an operation through simulation without actually performing that operation. ...

. *

CPLEX IBM ILOG CPLEX Optimization Studio (often informally referred to simply as CPLEX) is an optimization software package. In 2004, the work on CPLEX earned the first INFORMS Impact Prize. History The CPLEX Optimizer was named for the simplex ...

– integer, linear and quadratic programming. *

FEATool Multiphysics FEATool Multiphysics ("Finite Element Analysis Toolbox for Multiphysics") is a physics, finite element analysis (FEA), and PDE simulation toolbox. FEATool Multiphysics features the ability to model fully coupled heat transfer, fluid dynamics, ch ...

– FEA GUI Toolbox for MATLAB *

FICO Xpress The FICO Xpress optimizer is a commercial optimization solver for linear programming (LP), mixed integer linear programming (MILP), convex quadratic programming (QP), convex quadratically constrained quadratic programming (QCQP), second-order con ...

– integer, linear and quadratic and nonlinear programming. *

FortMP FortMP is a software package for solving large-scale optimization problems. It solves linear programming Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost ...

– integer, linear and quadratic programming.
FortranCalculus Compiler
– higher level language for Continuous Modeling, Simulation and Optimization. Solves Constrained, Implicit, Non-Linear, Algebraic and/or Ordinary Differential Equations using Solvers stored in a library. Most solvers use

Automatic Differentiation In mathematics and computer algebra, automatic differentiation (AD), also called algorithmic differentiation, computational differentiation, auto-differentiation, or simply autodiff, is a set of techniques to evaluate the derivative of a function ...

to calculate the Jacobian or Hessian matrix. *

FortSP FortSP is a software package for solving stochastic programming (SP) problems. It solves scenario-based SP problems with recourse as well as problems with chance constraints and integrated chance constraints. FortSP is available as a standalone ...

– stochastic programming. *

GAMS Gams is a municipality in the ''Wahlkreis'' (constituency) of Werdenberg in the canton of St. Gallen in Switzerland. History Gams is first mentioned in 835 as ''Campesias''. In 1210 it was mentioned as ''Chames'', in 1236 as ''Gamps''. Unt ...

– General Algebraic Modeling System. * :de:Gurobi – supports a range of optimization paradigms. * HEEDS MDO – multidisciplinary design optimization using SHERPA, a hybrid, adaptive optimization algorithm. *

IMSL Numerical Libraries IMSL (International Mathematics and Statistics Library) is a commercial collection of software libraries of numerical analysis functionality that are implemented in the computer programming languages C, Java, C#.NET, and Fortran. A Python inter ...

– linear, quadratic, nonlinear, and sparse QP and LP optimization algorithms implemented in standard programming languages C, Java, C# .NET, Fortran, and Python. *

IOSO IOSO (Indirect Optimization on the basis of Self-Organization) is a multiobjective, multidimensional nonlinear optimization technology. IOSO approach IOSO Technology is based on the response surface methodology approach. At each IOSO iteration the ...

– (Indirect Optimization on the basis of Self-Organization) a multiobjective, multidimensional nonlinear optimization technology. * Kimeme – an open platform for multi-objective optimization and multidisciplinary design optimization. *

LINDO Lindo is a surname. Notable people with the surname include: * Abigail Lindo (1803–1848), British lexicographer * Allan Lindo, more commonly known as apl.de.ap (born 1974), Filipino-American musician * Dean Lindo (born 1932), Belizean attorney * ...

- (Linear, Interactive, and Discrete Optimizer) a software package for linear programming, integer programming,

nonlinear programming In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints or the objective function are nonlinear. An optimization problem is one of calculation of the extrema (maxima, minima or st ...

, 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 Analytics is the systematic computational analysis of data or statistics. It is used for the discovery, interpretation, and communication of meaningful patterns in data. It also entails applying data patterns toward effective decision-making. It ...

modeling A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...

Learning and Intelligent OptimizatioN and reactive

business intelligence Business intelligence (BI) comprises the strategies and technologies used by enterprises for the data analysis and management of business information. Common functions of business intelligence technologies include reporting, online analytical pr ...

approach. *

modeFRONTIER modeFRONTIER is a multidisciplinary design optimization (MDO) platform developed by the Italian software house ESTECO SpA. Its workflow based environment, and multi-objective optimization algorithms are used for streamlining the engineering design ...

– an integration platform for multi-objective and multi-disciplinary 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 ''Acer'' () is a genus of trees and shrubs commonly known as maples. The genus is placed in the family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated since ht ...

– linear, quadratic, and nonlinear, continuous and integer optimization. Constrained and unconstrained. Global optimization with add-on toolbox. *

MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementa ...

– 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 MIDACO ( Mixed Integer Distributed Ant Colony Optimization) is a software package for numerical optimization based on evolutionary computing. MIDACO was created in collaboration of European Space Agency and EADS Astrium to solve constrained mixed ...

a lightweight software tool for single- and multi-objective

based on

evolutionary computing In computer science, evolutionary computation is a family of algorithms for global optimization inspired by biological evolution, and the subfield of artificial intelligence and soft computing studying these algorithms. In technical terms, they ...

. Written in C/C++ and Fortran with gateways to Excel, VBA, Java, Python, Matlab, Octave, R, C# and Julia. *

Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimi ...

– large-scale multivariate constrained and unconstrained, linear, quadratic and nonlinear, continuous and integer optimization. *

ModelCenter ModelCenter, developed by Phoenix Integration, is a software package that aids in the design and optimization of systems. It enables users to conduct trade studies, as well as optimize designs. It interfaces with other popular modeling tools, i ...

– a graphical environment for integration, automation, and design optimization. *

MOSEK MOSEK is a software package for the solution of linear, mixed-integer linear, quadratic, mixed-integer quadratic, quadratically constraint, conic and convex nonlinear mathematical optimization problems. The applicability of the solver varies wide ...

– 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 optimization; continuous or integer problems. *

NMath NMath is a numerical package for the Microsoft .NET Framework. It is developed by CenterSpace Software. Version 1.0 was released in March, 2003 as NMath Core. The current version is called NMath 7.1, released in December, 2019. NMath is built on ...

– linear, quadratic and nonlinear programming. *

Octeract Engine Octeract Engine is a proprietary deterministic global optimization solver for general Mixed-Integer Nonlinear Programs (MINLP).. It claims to use MPI as a means of accelerating solution times. Up to now there hasn't been any publication supporti ...

– a deterministic global optimization MINLP solver. Octeract Engine Community is free to use, and paid plans exist for additional features. *

OptimJ OptimJ is an extension for Java with language support for writing optimization models and abstractions for bulk data processing. The extensions and the proprietary product implementing the extensions were developed by Ateji which went out of busine ...

– Java-based modeling language. Premium Edition includes support for Mosek and CPLEX solvers. *

Optimus platform Optimus is a Process Integration and Design Optimization (PIDO) platform developed by Noesis Solutions. Noesis Solutions takes part in key research projects, such as PHAROS and MATRIX. Optimus allows the integration of multiple engineering ...

– a process integration and design optimization platform developed by Noesis Solutions. *

optiSLang optiSLang is a software platform for CAE-based sensitivity analysis, multi-disciplinary optimization (MDO) and robustness evaluation. It is developed by Dynardo GmbH and provides a framework for numerical Robust Design Optimization (RDO) and s ...

– software solutions for CAE-based sensitivity analysis, optimization and robustness evaluation. *

OptiY OptiY is a design environment software that provides modern optimization strategies and state of the art probabilistic algorithms for uncertainty, reliability, robustness, sensitivity analysis, data-mining and meta-modeling. Features OptiY is ...

- a design environment providing modern optimization strategies and state of the art probabilistic algorithms for uncertainty, reliability, robustness, sensitivity analysis, data-mining and meta-modeling. *

OptiStruct Altair Engineering Inc. is an American multinational information technology company headquartered in Troy, Michigan. It provides software and cloud solutions for simulation, IoT, high performance computing (HPC), data analytics, and artificia ...

– award-winning CAE technology for conceptual design synthesis and structural optimization. *

PottersWheel PottersWheel is a MATLAB toolbox for mathematical modeling of time-dependent dynamical systems that can be expressed as chemical reaction networks or ordinary differential equations (ODEs). It allows the automatic calibration of model parameters ...

– parameter estimation in ordinary differential equations (MATLAB toolbox, free for academic use). *

pSeven pSeven is a DSE (Design Space Exploration) software platform developed by DATADVANCE, extending design, simulation and analysis capabilities and assisting in faster design decisions. It provides integration with third party CAD and CAE software ...

— software platform for automation of engineering simulation and analysis, multidisciplinary optimization and data mining, developed by

DATADVANCE DATADVANCE Is a software development company, evolved out of a collaborative research program between Airbus and Institute for Information Transmission Problems of the Russian Academy of Sciences (IITP RAS). Product pSeven Core, embedded in ...

. * 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 SmartDO is a multidisciplinary design optimization software, based on the Direct Global Search technology developed and marketed by FEA-Opt Technology. SmartDO specialized in the CAE-Based optimization, such as CAE ( computer-aided engineering) ...

– multidisciplinary global design optimization, specialized in computer-aided engineering (CAE). using the direct global search approaches. *

SNOPT 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++, Py ...

– large-scale optimization problems. *

The Unscrambler The Unscrambler X is a commercial software product for multivariate data analysis, used for calibration of multivariate data which is often in the application of analytical data such as near infrared spectroscopy and Raman spectroscopy, and deve ...

X – product formulation and process optimization software. *

TOMLAB Tomlab is a German record label based in Köln. It has released works by bands such as The Books, Casiotone for the Painfully Alone, Deerhoof, Thee Oh Sees, Les Georges Leningrad, and acts associated with Blocks Recording Club, such as Final F ...

– supports global optimization, integer programming, all types of least squares, linear, quadratic and unconstrained programming for

. TOMLAB supports solvers like

KNITRO Artelys Knitro is a commercial software package for solving large scale nonlinear mathematical optimization problems. KNITRO – (the original solver name) short for "Nonlinear Interior point Trust Region Optimization" (the "K" is silent) – was ...

and

. *

VisSim VisSim is a visual block diagram program for simulation of dynamical systems and model-based design of embedded systems, with its own visual language. It is developed by Visual Solutions of Westford, Massachusetts. Visual Solutions was acq ...

– a visual

block diagram A block diagram is a diagram of a system in which the principal parts or functions are represented by blocks connected by lines that show the relationships of the blocks.

language for simulation and optimization of

dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...

s. *

WORHP WORHP ( "warp", an acronym for "We Optimize Really Huge Problems"), also referred to as eNLP (European NLP solver) by ESA, is a mathematical software library for numerically solving large scale continuous nonlinear optimization problems. WO ...

– a large-scale sparse solver for continuous nonlinear optimization.

Freeware/free for academic use

– free for academic and commercial use alike, with

Julia Julia is usually a feminine given name. It is a Latinate feminine form of the name Julio and Julius. (For further details on etymology, see the Wiktionary entry "Julius".) The given name ''Julia'' had been in use throughout Late Antiquity (e ...

, Python and

integrations. * ASTOS *

* Couenne – An open source solver for the deterministic global optimization of MINLPs licensed under the Eclipse Public License. *

Galahad library The Galahad library is a thread-safe library of packages for the solution of mathematical optimization problems. The areas covered by the library are unconstrained and bound-constrained optimization, quadratic programming, nonlinear programming, s ...

* GEKKO Python * LIONsolver *

– a software package for numerical

based on

. * MINTO –

integer programming An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective ...

solver using branch and bound algorithm; freeware for personal use. *

– a large scale optimization software. Solves linear, quadratic, conic and convex nonlinear, continuous and integer optimization. *

– Java-based modeling language; the free edition includes support for lp_solve,

GLPK The GNU Linear Programming Kit (GLPK) is a software package intended for solving large-scale linear programming (LP), mixed integer programming (MIP), and other related problems. It is a set of routines written in ANSI C and organized in the form ...

and LP or

MPS MPS, M.P.S., MPs, or mps may refer to: Science and technology * Mucopolysaccharidosis, genetic lysosomal storage disorder * Mononuclear phagocyte system, cells in mammalian biology * Myofascial pain syndrome * Metallopanstimulin * Potassium perox ...

file formats. *

– parameter estimation in ordinary differential equations (free MATLAB toolbox for academic use). *

Pyomo Pyomo is a collection of Python software packages for formulating optimization models. Pyomo was developed by William Hart and Jean-Paul Watson at Sandia National Laboratories and David Woodruff at University of California, Davis. Significant ...

- collection of Python software packages for formulating optimization models. *

References