Interior‐point method for non‐linear non‐convex optimization

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Abstract
In this paper, we propose an algorithm for solving non‐linear non‐convex programming problems, which is based on the interior point approach. Main theoretical results concern direction determination and step‐length selection. We split inequality constraints into active and inactive to overcome problems with stability. Inactive constraints are eliminated directly while active constraints are used to define symmetric indefinite linear system. Inexact solution of this system is obtained iteratively using indefinitely preconditioned conjugate gradient method. Theorems confirming efficiency of several indefinite preconditioners are proved. Furthermore, new merit function is defined, which includes effect of possible regularization. This regularization can be used to overcome problems with near linear dependence of active constraints. The algorithm was implemented in the interactive system for universal functional optimization UFO. Results of extensive numerical experiments are reported. Copyright © 2004 John Wiley & Sons, Ltd.

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