Quadratic programming methods for tailored reduced Hessian SQP

Reduced Hessian Successive Quadratic Programming (SQP) has been shown to be well suited for the solution of large-scale process optimization problems with many variables and constraints but only few degrees of freedom. The reduced space method Involves four major steps: an Initial preprocessing phase followed by an iterative procedure which requires the solution of a set of linear equalities, the QP subproblem and a line search. Clearly, the overall performance of the algorithm will depend directly on the robustness and computational efficiency of the techniques used to handle each of these subtasks. The preprocessing phase solves a linear feasibility problem corresponding to the original nonlinear programming formulation. This step serves to determine an initial consistent point, to select a nonsingular set of basis variables (required for the definition of the search subspaces) and to identify linear dependency among the equality constraints. We demonstrate how Fourer's piecewise-llnear simplex techniques allow us to solve a smaller initialization problem more efficiently than is possible with standard simplex techniques. Also, while the relative number of degrees of freedom is generally small, the actual number may become large. In this context, we present a new QP solver, QPKWIK, based on the dual algorithm of Goldfarb and Idnani. A unique feature of this algorithm is that it only requires the inverse Cholesky factor of the Hessian matrix to be supplied. At each iteration, this inverse Cholesky factor is obtained directly using a factorized inverse BFGS formula. The resulting solution technique for the QP subproblem is O(n) with respect to the degrees of freedom of the problem, as opposed to most existing software which involves O(n) operations. Further, the unconstrained optimum is dual feasible, which precludes the need for phase I calculations, and makes this method superior even for problems with few degrees of freedom. QPKWIK has been implemented so as to enhance the efficiency of the active set identification and is also able to determine a search direction when infeasible QP subproblems are encountered by relaxing the equality constraints without violating the simple bounds on the variables. Finally, numerical results are included, both to illustrate the advantages of the proposed techniques and to assess the overall performance of the reduced Hessian method. We note that this approach is especially wellsuited for process and real-time optimization and demonstrate this on several distillation problems as well as the Sunoco Hydrocracker Fractionation problem.