A practical Trust-Region SQP algorithm for equality- and bound-constrained optimization without derivatives

Abstract

In the last few years, a number of derivative-free optimization methods have been developed and especially model-based trust-region methods have been shown to perform well. Here, we present a new interpolation-based trust-region algorithm which can handle nonlinear and nonconvex optimization problems involving equality constraints and simple bounds on the variables. Our new algorithm is an extension of the algorithm BCDFO which handles bound constraints by an active-set method and has shown to be very competitive. It relies also on the technique of self-correcting geometry proposed by Scheinberg and Toint. The objective and constraint functions are approximated by polynomials of varying degree (linear or quadratic). The equality constraints are handled by a trust-region SQP approach, where each SQP step is decomposed into normal and tangential components. Special care must be taken in case an iterate is infeasible with respect to the models of the derivative-free constraints. Globalization is handled by using an Augmented Lagrangian penalty function as the merit function. We present numerical results on a test set of equality-constrained problems from the CUTEr problem collection and on a real-life application from engineering design in space craft development.