Given a transformation between input and output values, described by a mathematical function f, optimization 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 this case a minimization problem, can be represented in the following way
- Given: a function f : A R from some set A to the real numbers
- Search for: an element x0 in A such that f(x0) ≤ f(x) for all x in A.
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 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 optimization software organized according to license and business model type.
Free and open source software
Proprietary software
- AIMMS â€" optimization modeling system, including GUI building facilities
- Altair HyperStudy â€" Multi-Disciplinary Design of Experiments, Optimization and Stochastic study software
- AMPL â€" modelling language for large-scale linear, mixed integer and nonlinear optimization
- ANSYS DesignXplorer - Design exploration and optimization capabilities for the ANSYS Workbench platform
- APMonitor â€" Modeling language and optimization suite for large-scale, nonlinear, mixed integer, differential and algebraic equations with interfaces to MATLAB, Python, and Julia
- Artelys Knitro â€" large scale nonlinear optimization for continuous and mixed-integer programming
- ASTOS â€" AeroSpace Trajectory Optimization Software for launcher, re-entry and generic aerospace problems
- BARON â€" for global optimization of algebraic nonlinear and mixed-integer nonlinear problems
- COMSOL â€" a multiphysics simulation software with an add-on optimization module that includes several gradient-free and gradient-based optimizers, such as SNOPT, COBYLA, BOBYQA, Neld-Mead, GCMMA, etc.
- CPLEX â€" integer, linear and quadratic programming
- FortMP â€" integer, linear and quadratic programming
- FortSP â€" stochastic programming
- GAMS â€" General Algebraic Modeling System
- Gurobi â€" integer, linear and quadratic programming
- HEEDS MDO â€" 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 â€" (Indirect Optimization on the basis of Self-Organization) is a multiobjective, multidimensional nonlinear optimization technology
- Kimeme â€" 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 â€" an integrated software for data mining, modeling, Learning and Intelligent OptimizatioN and reactive business intelligence
- modeFRONTIER â€" 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 (software) â€" Linear, quadratic, and nonlinear, continuous and integer optimization. Constrained and unconstrained. Global optimization with add-on toolbox.
- MATLAB â€" Linear, integer, quadratic, and nonlinear problems with Optimization Toolbox; multiple maxima, multiple minima, and non-smooth optimization problems; estimation and optimization of model parameters.
- Mathematica â€" large-scale multivariate constrained and unconstrained, linear and nonlinear, continuous and integer optimization.
- MIDACO Global optimization software based on evolutionary computing, MINLP, parallelization, large-scale ( Matlab, Python, C/C++, Fortran, Java, C#, R, Java, VBA, Excel ).
- ModelCenter â€" a graphical environment for integration, automation, and design optimization.
- MOSEK â€" linear, quadratic, conic and convex nonlinear, continuous and integer optimization.
- NAG â€" 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 â€" linear, quadratic and nonlinear programming
- IOSO â€" multidisciplinary design optimization, for computer-aided engineering
- OptimJ â€" Java based modeling language. Premium Edition includes support for Gurobi, Mosek and CPLEX solvers
- Optimus platform â€" 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-modeling.
- OptiStruct â€" Award-winning CAE technology for conceptual design synthesis and structural optimization.
- PottersWheel â€" Parameter estimation in ordinary differential equations (Matlab toolbox, free for academic use).
- pSeven â€" software platform for automation of engineering simulation and analysis, multidisciplinary optimization and data mining, developed by DATADVANCE.
- SmartDO â€" multidisciplinary global design optimization, specialized in computer-aided engineering (CAE). using the direct global search approaches.
- SNOPT â€" large-scale optimization problems.
- Unscrambler Optimizer product formulation and process optimization software
- TOMLAB â€" TOMLAB supports global optimization, integer programming, all types of least squares, linear, quadratic and unconstrained programming for MATLAB. TOMLAB supports solvers like Gurobi, CPLEX, SNOPT, KNITRO and MIDACO.
- VisSim â€" a visual block diagram language for simulation and optimization of dynamical systems.
- WORHP â€" a large-scale sparse solver for continuous nonlinear optimization.
- XPRESS â€" integer, linear and quadratic and nonlinear programming
Freeware/Free for academic use
- AIMMS â€" free for academic use.
- AMPL â€" free for academic use.
- APMonitor â€" free for academic or commercial use with interfaces to Python and MATLAB.
- ASTOS â€" free for academic use.
- CPLEX â€" free for academic use.
- Galahad library â€" free for academic use.
- GUROBI â€" free for academic use.
- LIONsolver â€" free license for academic use.
- MIDACO Global optimization software based on evolutionary computing, MINLP, parallelization, large-scale ( Matlab, Python, C/C++, Fortran, Java, C#, R, Java, VBA, Excel ).
- MINTO â€" integer programming solver using branch and bound algorithm. Freeware for personal use.
- MOSEK â€" free for academic use.
- OptimJ â€" Java based modeling language. Free Edition includes support for lp_solve, GLPK and LP or MPS file formats
- PottersWheel â€" Parameter estimation in ordinary differential equations (Free Matlab toolbox for academic use).
- SCIP â€" Free only for research purpose as a member of a non-commercial and academic institution.
- WORHP â€" free of charge for academic use.
- XPRESS â€" free for academic use.
See also
- Comparison of optimization software
- List of numerical libraries
- List of computer algebra systems
- List of SMT solvers
- List of constraint programming languages
- List of optimization algorithms