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Given a system transforming a set of inputs to output values, described by a mathematical function f, optimization refers to the generation and selection of the best solution from some set of available alternatives, by systematically choosing input values from within an allowed set, computing the value of the function, and recording the best value found during the process. Many real-world and theoretical problems may be modeled in this general framework. For example, the inputs can be design parameters of a motor while the output can be the power consumption. Other inputs can be business choices with the output being obtained profit. or describing the configuration of a physical system with the output being its energy.
An optimization 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 ("minimization").
R from some Typically, A is some subset of the Euclidean space R, often specified by a set of constraints, equalities or inequalities that the members of A have to satisfy. Maximization can be reduced to minimization by multiplying the function by minus one.
The use of optimization software requires that the function f is defined in a suitable programming language and linked 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). 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 comparison of notable optimization software libraries, either specialized or general purpose libraries with significant optimization coverage.
| Name | Programming language | Latest stable version | Academic/noncommercial use is free |
Can be used in proprietary apps |
License | Notes |
|---|---|---|---|---|---|---|
| ALGLIB | C++, C#, Python, FreePascal | 3.19.0 / June 2022 | Yes | Yes | Dual (Commercial, GPL) | General purpose library, includes optimization package: linear, quadratic, nonlinear programming. |
| AMPL | C, C++, C#, Python, Java, Matlab, R | October 2018 | Yes | Yes | Dual (Commercial, academic) | A popular algebraic modeling language for linear, mixed-integer and nonlinear optimization. Student and AMPL for courses versions are available for free. |
| APMonitor | Fortran, C++, Python, Matlab, Julia | 0.6.2 / March 2016 | Yes | Yes | Dual (Commercial, academic) | A differential and algebraic modeling language for mixed-integer and nonlinear optimization. Freely available interfaces for Matlab, Python, and Julia. |
| Artelys Knitro | C, C++, C#, Python, Java, Julia, Matlab, R | 11.1 / November 2018 | No | Yes | Commercial, Academic, Trial | General purpose library, specialized in nonlinear optimization. Handles mixed-integer problems (MINLP) and mathematical programs with equilibrium constraints (MPEC). Specialized algorithms for nonlinear least squares problems. |
| CPLEX | C, C++, Java, C#, Python, R | 20.1 / Dec 2020 | Yes | Yes | Commercial, academic, trial | IBM CPLEX Optimization Studio is a suite of optimization engines (CPLEX for Mathematical Programming, and CP Optimizer for Constraint programming), a modeling language (OPL), and an Integrated Development Environment. |
| FICO Xpress | Mosel, BCL, C, C++, Java, R, Python, Matlab, .Net, VB6 | 8.13 / Nov 2021 | Yes | Yes | Commercial, academic, community, trial | Suite of Optimization Technologies and Solutions. Includes: Solver technologies including (LP (Simplex & Barrier), MIP, MIQP, MIQCQP, MISOCP, MINLP QP, QCQP, SOCP, NLP (SLP & Interior Point); An algebraic modelling and procedural programming language; an Integrated Development Environment; Supports for a range of execution services; Support for packaging of optimization models and services as software solutions |
| GEKKO | Python | 0.2.8 / August 2020 | Yes | Yes | Dual (Commercial, academic) | GEKKO is a Python package for machine learning and optimization of mixed-integer and differential algebraic equations. It is coupled with large-scale solvers for linear, quadratic, nonlinear, and mixed integer programming (LP, QP, NLP, MILP, MINLP). Modes of operation include parameter regression, data reconciliation, real-time optimization, dynamic simulation, and nonlinear predictive control. |
| GNU Linear Programming Kit | C | 4.52 / July 2013 | Yes | No | GPL | Free library for linear programming (LP) and mixed integer programming (MIP). |
| GNU Scientific Library | C | 1.16 / July 2013 | Yes | No | GPL | Free library provided by GNU project. |
| IMSL Numerical Libraries | C, Java, C#, Fortran, Python | many components | No | Yes | Proprietary | |
| LIONsolver | C++, Java | 2.0.198 / October 2011 | Yes | Yes | Proprietary | Support for interactive and learning optimization,
according to RSO principles . |
| Math Kernel Library (MKL) | C++, Fortran | 11.1 / October 2013 | No | Yes | Proprietary | Numerical library from Intel. MKL is specialized on linear algebra, but contains some optimization-related functionality. |
| Wolfram Mathematica | C++, Wolfram Language | 14.1.0 (July 31, 2024; 5 months ago (2024-07-31)) [±] | No | Yes | Proprietary | Constrained nonlinear optimization, interior point methods, convex optimization and integer programming-as well as original symbolic methods integrated with general computational capabilities. |
| MIDACO | C++, C#, Python, Matlab, Octave, Fortran, R, Java, Excel, VBA, Julia | 6.0 / Mar 2018 | Yes | Yes | Dual (Commercial, academic) | Lightweight software tool for single- and multi-objective optimization. Supporting MINLP and parallelization. |
| NAG Numerical Libraries | C, Fortran | Mark 26 / October 2017 | No | Yes | Proprietary | |
| NMath | C# | 5.3 / May 2013 | No | Yes | Proprietary | C# numerical library built on top of MKL. |
| Octeract Engine | C++/Python | 0.11.29 / November 2019 | No | Yes | Commercial | Supercomputing deterministic global optimization solver for general MINLP problems. Octeract Engine uses MPI for distributed calculations. |
| OptaPlanner | Java | 8.0.0.Final / November 2020 | Yes | Yes | ASL (open source) | Lightweight optimization solver in Java, with optional integration modules for JPA-Hibernate, Quarkus, Spring, Jackson, JAXB, etc. Works on Kotlin and Scala too. |
| SciPy | Python | 0.13.1 / November 2013 | Yes | Yes | BSD | General purpose numerical and scientific computing library for Python. |
See also
References
- "The Nature of Mathematical Programming," Mathematical Programming Glossary, INFORMS Computing Society.
- Battiti, Roberto; Mauro Brunato; Franco Mascia (2008). Reactive Search and Intelligent Optimization. Springer Verlag. ISBN 978-0-387-09623-0.
- "Yet More New Ideas and New Functions: Launching Version 14.1 of Wolfram Language & Mathematica". Retrieved 2024-07-31.
External links
- OR/MS Today: 2013 Linear Programming Software Survey
- OR/MS Today: 1998 Nonlinear Programming Software Survey
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