Solving the Infeasible Trust-Region Problem Using Approximations

Abstract

The experience in the mathematical community has shown that more eectiv e algorithms can be obtained by the specic inclusion of the constraints (SQP type of algorithms) rather than by using a penalty function as in the augmented Lagrangian formulation. The presence of explicit constraints in the local problem bounded by the trust region, however, may have no feasible solution. In order to remedy this problem the mathematical community has developed dieren t versions of a composite steps approach. This approach consists of a normal step to reduce the amount of constraint violation and a tangential step to minimize the objective function maintaining the level of constraint violation attained at the normal step. Two of the authors have developed a dieren t approach for a sequential approximate optimization frame- work using homotopy ideas to relax the constraints. This algorithm called interior-point trust-region se- quential approximate optimization (IPTRSAO) presents some similarities to the two normal-tangential steps algorithms. In this paper, a description of the similarities is presented and an expansion of the two steps algorithm is presented for the case of approximations.