Nonconvex process optimization

Abstract

Difficulties associated with nonconvexity in successive quadratic programming (SQP) methods are studied. It is shown that projected indefiniteness of the Hessian matrix of the Lagrangian function can (i) place restrictions on the order in which inequalities can be added or deleted from the active set, (ii) generate redundant active sets whose resolution is nontrivial, (iii) give rise to quadratic programming (QP) subproblems that have multiple Kuhn-Tucker points, and (iv) produce nondescent directions in the SQP method that can lead to failure. Related issues concerned with the use of feasible or infeasible starting points for the iterative quadratic programs, forcing positive definiteness to ensure convexity, and using iterative methods to solve the linear Kuhn-Tucker conditions associated with the QP subproblems are also studied. A new active set strategy that (i) monitors projected indefiniteness to guide the addition of constraints to the active set, (ii) permits line searching for negative values of the line search parameter, and (iii) does not necessarily delete active constraints with incorrect Kuhn-Tucker multipliers is proposed. Constraint redundancy is circumvented using an algorithm that identifies all nontrivial redundant subsets of smallest size and determines which, if any, exchanges are justified. Nondescent in the NLP's is resolved using a linear programming (LP)-based trust region method that guarantees descent regardless of merit function. It is also shown that there is no justification for using feasible starting points at the QP level of the calculations, that forcing positive definiteness to ensure convexity can cause termination at undesired solutions, and that the use of iterative methods to solve the linear Kuhn-Tucker equations for the QP's can cause a deterioration in numerical performance. Many small chemical process examples are used to highlight difficulties so that geometric illustrations can be used while heat exchange network design and distillation operations examples are used to show that these same difficulties carry over to larger problems.