Reduced successive quadratic programming implementation for large-scale optimization problems with smaller degrees of freedom

Abstract

Many process optimization problems consist of large numbers of equations and only a few degrees of freedom. To exploit this feature an implementation of the range and null space decomposition (RND) strategy for large-scale problems is described, with emphasis on the optimization of engineering systems. The RND technique for successive quadratic programming (SQP), as detailed in Vasantharajan and Biegler ( Comput. chem. Engng 12, 1087–1101, 1988), is based on an inexpensive nonorthonormal decomposition. However, the original study evaluated a dense implementation. Here we discuss the incorporation of general-purpose sparse matrix techniques. Also, we consider safeguards for dealing with inconsistent linearizations and infeasible quadratic programs (QPs), which can compromise the robustness of this method. Here systematic ways of generating a nonsingular basis for general nonlinear programs are implemented to adapt this strategy to solve large, sparse problems efficiently. We describe an LP-based procedure which serves to partition the variables into decisions and dependents, thereby generating a nonsingular basis. Any redundancies/degeneracies in the constraints are also detected and processed separately. The reduced SQP implementation has been integrated into the modeling system, GAMS (Brooke et al., GAMS, A User's Guide, 1988). Finally, a comparison of the RND-based reduced SQP strategy with MINOS (Murtagh and Saunders, Minos/Augmented Supplementary User's Manual, 1978) is presented on a set of NLPs and process design problems. The process problems include the optimization of the operation of distillation columns. These problems warrant special mention as they have been implemented in a fully equation-oriented manner, thus exploiting the potential of the GAMS architecture. Results are presented that confirm the success of the reduced SQP implementation for efficient solution of large, difficult nonlinear programs.