Robust code for constrained optimization

Introduction E ASON and Fenton and Pappas have evaluated the performance of eighteen optimization codes for solving nonlinear programming problems. All codes were applied to ten problems having inequality constraints, and the results were used to rank the codes in terms of generality and efficiency. Of the eighteen codes, five used information concerning derivatives or derivative approximations; the best of these five codes solved only half of the problems under a specified set of test conditions. As a result, the gradient-based codes ranked low in terms of performance. Among the recent advances in gradient-based methods have been the development of a variety of augmented Lagrangian methods (also called multiplier methods) which were originally introduced' for equality constrained problems, but have since been generalized in various ways to account for inequality constraints as well. One of these generalizations led to the development of an algorithm and a FORTRAN code LPNLP which has proved to be very efficient in solving a variety of test problems and application problems; a method using gradient approximations with LPNLP has been evaluated in Ref. 8. In contrast to conventional gradient-based penalty function methods, the augmented Lagrangian methods do not require in general that penalty weights approach infinity as the solution progresses. The purpose of this Note is to present the results of applying LPNLP to the ten problems cited in Ref. 1. The results show that LPNLP compares favorably with the best of the codes tested in Refs. 1 and 3. Of interest are problems of the form