Nonconvexity and Descent in Nonlinear Programming

Abstract

Nonconvexity in nonlinear and quadratic programming is studied in the context of a full space successive quadratic programming (SQP) method with analytical second derivatives. It is shown that nonconvexity can lead to indefinite quadratic programs and multiple Kuhn-Tucker points in both the quadratic and nonlinear programs. It is also shown that some quadratic programming solutions are not descent directions for the parent SQP method and can lead to increased computational work or failure in the nonlinear programming calculations. A simple heuristic-based methodology is proposed to improve the chances of calculating a descent direction for the nonlinear programming algorithm. The associated algorithmic logic is based on pruning and mirroring. Several chemical process optimization problems are used to show that the proposed methodology is useful in calculating descent directions from indefinite quadratic programs and often results in convergence to the desired nonlinear programming solution in a reliable and efficient manner.