Non-monotone derivative-free algorithm for solving optimization models with linear constraints: extensions for solving nonlinearly constrained models via exact penalty methods

Abstract

This paper describes a non-monotone direct search method (NMDSM) that finds a stationary point of linearly constrained minimization problems. At each iteration the algorithm uses NMDSM techniques on the Euclidean space ${\mathbb {R}}^n$ spanned by n variables carefully selected from the $n+m$ variables formulated by the model under analysis. These variables are obtained by simple rules and are handled with pivot transformations frequently used in the solution of linear systems. A new weaker 0-order non smooth necessary condition is suggested, which transmute to other stationarity conditions, depending upon the kind of differentiability present in the system. Convergence with probability 1 is proved for non smooth functions. The algorithm is tested numerically on a set of small to medium size problems that have exhibited serious difficulties for their solution by other optimization techniques. The paper also considers possible extensions to non-linearly constrained problems via exact penalty function and a slightly modified algorithm satisfactorily solved a multi-batch multi-product plant that was modeled as a MINLP.