Nonlinear Symbolic Transformations for Simplifying Optimization Problems

Abstract

The theory of nonlinear optimization traditionally studies numeric computations. However, increasing attention is being paid to involve computer algebra into mathematical programming. One can identify two possibilities of applying symbolic techniques in this field. Computer algebra can help the modeling phase by producing alternate mathematical models via symbolic transformations. The present paper concentrates on this direction. On the other hand, modern nonlinear solvers use more and more information about the structure of the problem through the optimization process leading to hybrid symbolic-numeric nonlinear solvers. This paper presents a new implementation of a symbolic simplification algorithm for unconstrained nonlinear optimization problems. The program can automatically recognize helpful transformations of the mathematical model and detect implicit redundancy in the objective function. We report computational results obtained for standard global optimization test problems and for other artificially constructed instances. Our results show that a heuristic (multistart) numerical solver takes advantage of the automatically produced transformations. New theoretical results will also be presented, which help the underlying method to achieve more complicated transformations.