Nonconvexity in Chemical Process Optimization

摘要-LUIA-9312066理论上合理、可靠和高效的计算机工具，用于寻找大型化工过程优化问题的局部解，一直是PI研究的重点。他一直致力于开发一种求解非线性约束优化问题的连续二次规划(SQP)方法(此类问题的一个例子是精馏塔等分离装置的设计)。SQP方法是一类非常强大的求解非线性规划问题的方法，特别是那些具有强非线性约束的规划问题。SQP方法的一个最大缺点是迭代QP解可以给出非线性规划(NLP)的非下降方向，即使对于凸二次规划也是如此。对于这个研究项目，PI计划：a)开发基于线性规划的非对称信赖域方法的理论框架，当二次规划解不是下降方向时，该方法保证下降。B)进行相应的数值研究，旨在制定必要的规则来调整信赖域的大小，以保证下降和收敛到局部Kuhn-Tucker点。C)研究鞍点解对非线性规划算法的理论和计算意义。此外，他计划通过以下方式来研究非凸性在二次规划中的一般作用：1)对非凸性与当前有效集方法的不足、约束冗余的出现、多重性的存在、非降性、起点可行性/不可行性的要求、投影确定性、不确定性和投影奇异性之间的关系进行数值调查和研究。2)并行开发可靠和高效的算法来解决这些和其他问题，包括：(A)新的活动集方法，能够处理具有可行或不可行起点的投影不确定性；(B)预处理算法，可以识别给定约束集的所有最小非平凡冗余子集；以及(C)当遇到奇点时，将(B)并入(A)。3)为指导和支持上面2)中的算法发展奠定了理论基础。

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.