Nelder-Mead optimization under linear constraints

Abstract

An extension of the Nelder-Mead simplex algorithm is presented in this dissertation.  The algorithm was developed for minimizing a non-linear objective function subject to linear inequality constrains.  The algorithm assumes that the objective function is analytically unavailable or its evaluation at each experimental design point is very expensive.  The algorithm generates a feasible trial point at each iteration and compares it with the current pattern of points (simplex).  The algorithm, called the Linear Constraint Nelder-Mead (LCNM), takes advantage of when the current simplex is eventually flattened by a constrained reflection or expansion operation, as a result of meeting the boundary of the feasible region.  When this occurs, the algorithm reduces the number of vertices of the current simplex thereby avoiding the degeneration of the simplex.  A limited study of its performance is developed by case studies.  Although the LCNM algorithm was designed for minimizing linearly constrained non-linear objective functions, a particular case of linear programming is theoretically studied, showing that a very slow convergence rate is possible.  A modification to the LCNM algorithm was included for improving the convergence rate of the algorithm. Two variations of the algorithm were also investigated.  The LCNM algorithm has displayed a good enough performance when the objective function is corrupted by noise.  This fact allows us to appreciate the LCNM algorithm as an optimization method for problems of optimization by simulation, where the evaluation of the design points can require a huge computational effort.

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