Inexact-Restoration Algorithm for Constrained Optimization1

We introduce a new model algorithm for solving nonlinear programming problems. No slack variables are introduced for dealing with inequality constraints. Each iteration of the method proceeds in two phases. In the first phase, feasibility of the current iterate is improved; in second phase, the objective function value is reduced in an approximate feasible set. The point that results from the second phase is compared with the current point using a nonsmooth merit function that combines feasibility and optimality. This merit function includes a penalty parameter that changes between consecutive iterations. A suitable updating procedure for this penalty parameter is included by means of which it can be increased or decreased along consecutive iterations. The conditions for feasibility improvement at the first phase and for optimality improvement at the second phase are mild, and large-scale implementation of the resulting method is possible. We prove that, under suitable conditions, which do not include regularity or existence of second derivatives, all the limit points of an infinite sequence generated by the algorithm are feasible, and that a suitable optimality measure can be made as small as desired. The algorithm is implemented and tested against the LANCELOT algorithm using a set of hard-spheres problems.