A pattern search and implicit filtering algorithm for solving linearly constrained minimization problems with noisy objective functions

Abstract

PSIFA - Pattern Search and Implicit Filtering Algorithm - is a derivative-free algorithm that has been designed for linearly constrained problems with noise in the objective function. It combines some elements of the pattern search approach of Lewis and Torczon with ideas from the method of implicit filtering of Kelley enhanced with a further analysis of the current face and a simple extrapolation strategy for updating the step length. The feasible set is explored by PSIFA without any particular assumption about its description, being the equality constraints handled in their original formulation. Besides, compact bounds for the variables are not mandatory. The global convergence analysis is presented, encompassing the degenerate case, under mild assumptions. Numerical experiments with linearly constrained problems from the literature were performed. Additionally, problems with the feasible set defined by polyhedral 3D cones with several degrees of degeneration at the solution were addressed, including noisy functions that are not covered by the theoretical hypotheses. To put PSIFA in perspective, comparative tests have been prepared, with encouraging results