Global Optimization by Differential Evolution and Particle Swarm Methods Evaluation on Some Benchmark Functions

I. A Brief History of Optimization Research: The history of optimization of realvalued non-linear functions (including linear ones), unconstrained or constrained, goes back to Gottfried Leibniz, Isaac Newton, Leonhard Euler and Joseph Lagrange. However, those mathematicians often assumed differentiability of the optimand as well as constraint functions. Moreover, they often dealt with the equality constraints. Richard Valentine (1937) and William Karush (1939), however, were perhaps the first mathematicians to study optimization of nonlinear functions under inequality constraints. Leonid Kantorovich and George Dantzig are well known for developing and popularizing linear programming, which ushered a new era of ‘operations research’, a branch of mathematical science that specializes in optimization. The development of linear programming soon prompted the study of the optimization problem of nonlinear functions (often under linear or nonlinear constraints). The joint work of Harold Kuhn and Albert Tucker (1951) – that was backed up by the work of Karush – is a landmark in the history of optimization of nonlinear functions.