ABSTRACT

Mathematical programs with vanishing constraints (MPVCs) are a class of nonlinear optimization problems with applications to various engineering problems such as truss topology design and robot motion planning. MPVCs are difficult problems from both a theoretical and numerical perspective: the combinatorial nature of the vanishing constraints often prevents standard constraint qualifications and optimality conditions from being attained; moreover, the feasible set is inherently nonconvex, and often has no interior around points of interest. In this paper, we therefore study and compare four regularization methods for the numerical solution of MPVCs. Each method depends on a single regularization parameter, which is used to embed the original MPVC into a sequence of standard nonlinear programs. Convergence results for these methods based on both exact and approximate stationary of the subproblems are established under weak assumptions. The improved regularity of the subproblems is studied by providing sufficient conditions for the existence of KKT multipliers. Numerical experiments, based on applications in truss topology design and an optimal control problem from aerothermodynamics, complement the theoretical analysis and comparison of the regularization methods. The computational results highlight the benefit of using regularization over applying a standard solver directly, and they allow us to identify two promising regularization schemes.

摘要

具有消失约束的数学程序 (MPVC) 是一类非线性优化问题，可应用于各种工程问题，例如桁架拓扑设计和机器人运动规划。从理论和数值的角度来看，MPVC 都是难题：消失约束的组合性质通常会阻止获得标准约束条件和最优条件；此外，可行集本质上是非凸的，并且通常在兴趣点周围没有内部。因此，在本文中，我们研究和比较了 MPVC 数值解的四种正则化方法。每种方法都依赖于一个正则化参数，该参数用于将原始 MPVC 嵌入到一系列标准非线性程序中。基于子问题的精确和近似平稳的这些方法的收敛结果是在弱假设下建立的。通过为KKT乘数的存在提供充分条件来研究子问题的改进规律。数值实验基于桁架拓扑设计中的应用和空气热力学的最优控制问题，补充了正则化方法的理论分析和比较。计算结果突出了使用正则化比直接应用标准求解器的好处，它们使我们能够确定两种有前途的正则化方案。通过为KKT乘数的存在提供充分条件来研究子问题的改进规律。数值实验基于桁架拓扑设计中的应用和空气热力学的最优控制问题，补充了正则化方法的理论分析和比较。计算结果突出了使用正则化比直接应用标准求解器的好处，它们使我们能够确定两种有前途的正则化方案。通过为KKT乘数的存在提供充分条件来研究子问题的改进规律。数值实验基于桁架拓扑设计中的应用和空气热力学的最优控制问题，补充了正则化方法的理论分析和比较。计算结果突出了使用正则化比直接应用标准求解器的好处，它们使我们能够确定两种有前途的正则化方案。