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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*x*_{0}in*A*such that*f*(*x*_{0}) ≤*f*(*x*) for all*x*in*A*.

In continuous optimization, *A* is some subset of the Euclidean space **R**^{n}, 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, 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.

Name | License | Brief info |
---|---|---|

ADMB | BSD | nonlinear optimization framework, using automatic differentiation |

APMonitor | BSD | MATLAB Toolbox and Python APIs to Mixed Integer Nonlinear Programming Solvers |

ASCEND | GPL | mathematical modelling system |

CONDOR | GPL | Non-linear Continuous Objective Function for small dimension (n<20) with linear and non-linear constraints. Only the value of the objective function is used. Stand-Alone C++ code. |

COIN-OR SYMPHONY | GPL | integer programming |

CUTEr | GPL | testing environment for optimization and linear algebra solvers |

IPOPT | CPL | large scale nonlinear optimization for continuous system |

JOptimizer | ASL | open source java library for convex optimization |

OpenOpt | BSD | free numerical optimization framework for solving NLP, LP, MIP, QP, with automatic differentiation features. |

OpenSolver | GPL | open source Excel addin for solving LP and MIP problems |

OPTI Toolbox | BSD | MATLAB Toolbox for solving linear, nonlinear, continuous and discrete optimization problems. |

MOEA Framework | LGPL | open source Java library for multiobjective optimization |

UFFLP | GPL | open source. An easy API for Mixed, Integer and Linear Programming |

- AIMMS — optimization modeling system, including GUI building facilities
- AMPL — modelling language for large-scale linear, mixed integer and nonlinear optimization
- APMonitor — for large-scale, nonlinear, differential and algebraic optimization
- ASTOS — AeroSpace Trajectory Optimization Software for launcher, re-entry and generic aerospace problems
- BARON — for global optimization of algebraic nonlinear and mixed-integer nonlinear problems
- CPLEX — integer, linear and quadratic programming
- FortMP — integer, linear and quadratic programming
- FortSP — stochastic programming
- General Algebraic Modeling System
- Gurobi — integer, linear and quadratic programming
- 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) is a multiobjective, multidimensional nonlinear optimization technology
- Kimeme — an open platform for multi-objective optimization and multidisciplinary design optimization
- LIONsolver — an integrated software for data mining, modeling,
*Learning and Intelligent OptimizatioN*and reactive business intelligence - MIDACO-Solver — Black-Box optimization software for continuous, combinatorial and mixed integer problems. Available in Matlab, C/C++, Fortran and Python.
- modeFRONTIER — multidisciplinary design optimization, for computer-aided engineering
- Maple (software) — Linear, quadratic, and nonlinear, continuous and integer optimization. Constrained and unconstrained. Global optimization with add-on toolbox.
- MATLAB — Standard and large -scale optimization problems; multiple maxima, multiple minima, and non-smooth optimization problems; estimation and optimization of model parameters.
- Mathematica — large-scale multivariate constrained and unconstrained, linear and nonlinear, continuous and integer 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 optimization; continuous or integer problems.
- Nexus — multidisciplinary optimization and design process integration
- NMath — linear, quadratic and nonlinear programming
- IOSO — multidisciplinary design optimization, for computer-aided engineering
- OptimJ — Java based modeling language. Premium Edition includes support for Gurobi, Mosek and CPLEX solvers
- Optimus platform — a process integration and design optimization platform developed by Noesis Solutions
- 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 — Award-winning CAE technology for conceptual design synthesis and structural optimization.
- PHX ModelCenter — a graphical environment for automation, integration, and design optimization.
- PottersWheel — Parameter estimation in ordinary differential equations (Matlab toolbox, free for academic use).
- Sipesc.Opt — general purpose optimization software, including GUI and script language interface.
- SmartDO — multidisciplinary global design optimization, specialized in computer-aided engineering (CAE). using the direct global search approaches.
- SNOPT — large-scale optimization problems.
- Unscrambler Optimizer product formulation and process optimization software
- TOMLAB — 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 and KNITRO.
- VisSim — a visual block diagram language for simulation and optimization of dynamical systems.
- WORHP — a large-scale sparse solver for continuous nonlinear optimization.
- Xpress-Mosel by FICO — "ultra-large-scale" optimization with 64-bit support.

- AIMMS — free for academic use.
- AMPL — free student license.
- APMonitor — free for academic or commercial use with interfaces to Python and MATLAB.
- ASTOS — free for academic use.
- Galahad library — free for academic use.
- LIONsolver — free license for academic use.
- MIDACO-Solver — free limited version (Matlab, C/C++, Fortran, Python)
- MINTO — integer programming solver using branch and bound algorithm. Freeware for personal use.
- OptimJ — Java based modeling language. 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).
- Sipesc.Opt — free for academic use.
- SCIP — Free only for research purpose as a member of a non-commercial and academic institution.
- WORHP — free of charge for academic use.