一类互补约束优化问题的一个扰动方法的收敛性

On the convergence of a perturbation approach for a class of optimization problems with complementarity constraints

互补约束优化问题是一类重要的最优化问题,在科学和工程中有着重要的应用.交通规划的道路扩容问题,经济学领域的DICE模型都是互补约束优化问题.这类问题因为约束集合不满足通常的约束规范而不能用传统的非线性规划方法处理,往往用光滑近似的方法来克服这一困难.考虑一类互补约束优化问题的基于光滑化Fischer-Burmeister函数的扰动方法.证明了当光滑化参数μ↘0时扰动问题的值收敛到原问题的最优值,扰动问题的最优解集合的外极限包含在问题最优解集合中.说明扰动问题很容易满足通常的约束规范,并给出扰动问题的一阶必要性最优条件和二阶充分性最优条件.

Mathematical programs with complementarity constraints are an important class of optimi- zation problems, which have important applications in science and engineering. For examples, the road capacity expansion problem in transportation and the DICE model in economics are such kind of problems. Traditional nonlinear programming solvers can not be used to solve mathematical programs with complementarity constraints, because conventional constraint qualifications do not hold for the constraint sets, and smooth approximationmethods are proposed to overcome such a difficulty. This paper considers the perturbation approach based on the smoothed Fischer-Burmeister function for a class of optimization problems with complementarity constraints. We prove that the optimal value of the perturbed problem converges to that of the original problem and the outer limit of the solution set for the perturbed problem is contained in the solution set of the original problem when the smoothing parameter μ↓0. We explain why the conventionally used constraint qualifications are easily satisfied and present the first-order necessary optimality conditions and the second-order sufficient optimality conditions for the perturbed problems.