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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.
The SciPy scientific library, for instance, uses HiGHS as its LP solver [13] from release 1.6.0 [14] and the HiGHS MIP solver for discrete optimization from release 1.9.0. [15] As well as offering an interface to HiGHS, the JuMP modelling language for Julia [ 16 ] also describes the specific use of HiGHS in its user documentation. [ 17 ]
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.
GEKKO is an extension of the APMonitor Optimization Suite but has integrated the modeling and solution visualization directly within Python. A mathematical model is expressed in terms of variables and equations such as the Hock & Schittkowski Benchmark Problem #71 [ 2 ] used to test the performance of nonlinear programming solvers.
Xpress was originally developed by Dash Optimization, and was acquired by FICO in 2008. [3] Its initial authors were Bob Daniel and Robert Ashford. The first version of Xpress could only solve LPs; support for MIPs was added in 1986. Being released in 1983, Xpress was the first commercial LP and MIP solver running on PCs. [4]
For example, if the feasible region is defined by the constraint set {x ≥ 0, y ≥ 0}, then the problem of maximizing x + y has no optimum since any candidate solution can be improved upon by increasing x or y; yet if the problem is to minimize x + y, then there is an optimum (specifically at (x, y) = (0, 0)).
Dlib is a modern C++ library with easy to use linear algebra and optimization tools which benefit from optimized BLAS and LAPACK libraries. Eigen is a vector mathematics library with performance comparable with Intel's Math Kernel Library; Hermes Project: C++/Python library for rapid prototyping of space- and space-time adaptive hp-FEM solvers.
Here some test functions are presented with the aim of giving an idea about the different situations that optimization algorithms have to face when coping with these kinds of problems. In the first part, some objective functions for single-objective optimization cases are presented.