Sparse quadratic programming in chemical process optimization

Abstract

The quadratic programming aspects of a full space successive quadratic programming (SQP) method are described. In particular, fill-in, matrix factor and active set updating, numerical stability, and indefiniteness of the Hessian matrix are discussed in conjunction with a sparse modification of Bunch and Parlett factorization of symmetric indefinite (Kuhn-Tucker) matrices of the type often encountered in optimization. A new pivoting strategy, called constrained pivoting, is proposed to reduce fill-in and compared with complete, partial and threshold pivoting. It is shown that constrained pivoting often significantly reduces fill-in and thus the iterative computational burdens associated with the factorization and solution of Kuhn-Tucker conditions within the QP subproblem. A numerical algorithm for updating the lower triangular and diagonal factors is presented and shown to be very fast, usually requiring only about 5% of the cost of refactorization. Two active set strategies are also presented. These include the options of adding inequalities either one or several at a time. In either case, the effects on matrix factor updating is shown to be small. Finally, a simple test is used to maintain iterative descent directions in the quadratic program. Our sparse symmetric indefinite QP algorithm is tested in the context of a family of SQP algorithms that include a full space Newton method with analytical derivatives, a full space BFGS method and a Range and Null space Decomposition (RND) method in which the projected Hessian is calculated from either analytical second derivatives or the BFGS update. Several chemical process optimization problems, with small and large degrees of freedom, are used as test problems. These include minimum work calculations for multistage isothermal compression, minimum area targeting for heat exchanger networks, and distillation optimizations involving some azeotropic and extractive distillations. Numerical results show uniformly that both the proposed QP and SQP algorithms, particularly the full space Newton method, are reliable and efficient. No failures were experienced at either level.