Abstract - Lucia - 9312066 Theoretically sound and reasonably reliable and efficient computer tools for finding local solutions to large chemical process optimization problems have been the focus of this PI's research. He has been working on developing a successive quadratic programming (SQP) method for nonlinearly constrained optimizations problems (an example of such problems are the design of separation units such as distillation columns). SQP methods are a very powerful class of methods for solving nonlinear programming problems, particularly those with strong nonlinear constraints. The single biggest weakness of SQP methods is the fact that iterative QP solutions can give directions of nondescent in the nonlinear program (NLP), even for convex quadratic programs. For this research project the PI plans to: A) Develop the theoretical framework for linear programming-based, asymmetric trust region methods that guarantee descent when the quadratic programming solution is not a descent direction. B) Conduct concomitant numerical studies directed at developing the necessary rules for adjusting the trust region size to guarantee both descent and convergence to a local Kuhn-Tucker point. C) Study the theoretical and computational implications of saddlepoint solutions to nonlinear programming algorithms. In addition, he plans to look at the general role of nonconvexity in quadratic programming by: 1) Conducting a numerical investigation and study of the relationship between nonconvexity and the inadequacy of current active set methods, the occurrence of constraint redundancy, the presence of multiplicity, nondescent, the requirement of starting point feasibility/infeasibility, projected definiteness, indefiniteness, and projected singularity. 2) Developing, in parallel, reliable and efficient algorithms that address these and other issues, including (a) new active set methods capable of handling projected indefiniteness with feasible or infeasible starting points, (b) preprocessing a lgorithms that can identify all smallest nontrivial redundant subsets of a given set of constrains, and (c) incorporating (b) into (a) when singularity is encountered. 3) Establishing theoretical foundations to both guide and support the algorithmic developments in 2) above.
