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Constrained optimization problems (COPs) are very important in that they frequently appear in the real world. A COP, in which both the function and constraints may be nonlinear, consists of the optimization of a function subject to constraints. Constraint handling is one of the major concerns when solving COPs with particle swarm optimization (PSO) combined with the Nelder-Mead simplex search method (NM-PSO). This article proposes embedded constraint handling methods, which include the gradient repair method and constraint fitness priority-based ranking method, as a special operator in NM-PSO for dealing with constraints. Experiments using 13 benchmark problems are explained and the NM-PSO results are compared with the best known solutions reported in the literature. Comparison with three different meta-heuristics demonstrates that NM-PSO with the embedded constraint operator is extremely effective and efficient at locating optimal solutions. [ABSTRACT FROM AUTHOR]

There is abundant literature on response surface methodology about how the Nelder–Mead simplex search (NMSS) procedure can be applied to the determination of optimum operating conditions for deterministic response surface functions. However, searching for the optima of stochastic functions seems a more realistic task for many practical situations. This is particularly true when the response surface functions have not been fitted or the physical models may not even exist, so gradient information is not available. In such cases, it might be interesting to employ simplex-search-type methods ‘on-line’ to sequentially optimize the actual response of interest. Toward that end, an enhanced NMSS is proposed in this article to explore the terrains of empirical (or experimental) optimization adaptively where the known response surface function incorporates additional white-noise errors. Internal modifications to basic operations in NMSS are made primarily according to some statistical process control statistics in estimating response variation and confidence bands for mean responses. A series of graphical illustrations are presented to give an insight into the way the new simplex-search-type approach accurately anchors the true optimum point in noisy environments. As evidenced by a wide variety of simulation studies on the published response functions, the new method proves to perform much better than two recent modifications of NMSS in solution quality achieved while applied to the stochastic response surface optimization problems. [ABSTRACT FROM AUTHOR]