均衡约束优化问题的若干研究

发布时间：2018-06-03 12:44

本文选题：均衡约束数学规划 + 同伦方法　；参考：《吉林大学》2016年博士论文

【摘要】：均衡约束数学规划问题(Mathematical Programs with Equilibrium Constraints,简称MPEC)是指约束集中含有参数变分不等式、互补问题和广义方程的约束规划问题.该问题广泛应用于数理经济,工程设计,化学工程,交通科学等领域,并且与变分不等式问题、Nash均衡、互补问题等有着紧密的联系.然而由于MPEC问题的可行域不满足大部分的约束规范,尤其是M-F约束规范(Magasarian-Fromvitz),在可行域的任意一点处都不满足,所以这类问题在理论分析和算法求解中都是非常困难的.在过去的二十多年里,关于MPEC问题在理论和算法方面的研究取得了丰硕的成果,但是仍有许多问题有待于解决.在本文中,我们基于投影函数和光滑化理论,利用组合同伦算法深入研究了几类带有均衡约束的数学规划问题和均衡约束多目标优化问题,主要取得了以下成果.1、研究带有有界箱式约束变分不等式的数学规划问题.首先将所求问题中的变分不等式价转化为带有投影函数的非光滑等式,再基于箱式约束集合的特点,利用Cabriel-More光滑函数逼近等式中的非光滑部分,构造一个带参数的等式约束,对转化后的数学规划问题的KKT系统构造同伦方程.这种做法既不需要假设函数F具有强单调性,也不需要引入额外的变量,而且在后续的计算中方便了初始点的选取.最后证明了同伦路径的存在性和大范围的收敛性,并通过数值实验验证了算法的可行性和有效性.2、建立了互补约束数学规划问题的同伦算法.首先将带有互补约束的数学规划问题转化为一般的非光滑函数约束的非线性规划问题.然后利用光滑化手段把其中的非光滑等式约束转化为光滑函数的等式约束.从而将前述的数学规划问题转化为光滑函数约束的数学规划问题,这样避免了引入更多的乘子变量.对最后得到的光滑规划问题的KKT系统构造同伦方程,证明了同伦路径的存在性和收敛性,同时证明了所求得到的KKT点是原问题的C-稳定点,并且利用数值算例验证了算法的可行性和有效性.3、构建了求解带有均衡约束的多目标优化问题的新的同伦算法.首先利用SBCQ约束规范将原问题等价转化带有KKT系统的一般的多目标问题,将上述的KKT系统转化为一个光滑的等式约束,进而得到一个带有等式和不等式约束的多目标规划问题,最后对转化后的等价问题的KKT系统构造同伦方程,证明了同伦路径地存在性和收敛性.最后用数值实验证明了所提出的算法的可行性和有效性.4、讨论了约束条件中变分不等式定义在一般闭凸集上的均衡约束规划问题.通过引入无穷远解的概念,将所求的MPEC问题转化为带有投影函数的单层优化问题。再利用光滑化手段将最后得到的单层优化问题转化为光滑函数约束的优化问题,对其KKT系统构造同伦方程,证明了同伦路径的存在性和收敛性,并给出计算实例.
[Abstract]:Equilibrium constrained Programs with Equilibrium Constraints, (MPECs) is a constrained programming problem with parametric variational inequalities, complementary problems and generalized equations in the constraint set. This problem is widely used in the fields of mathematical economics, engineering design, chemical engineering, traffic science and so on, and is closely related to the variational inequality problems such as Nash equilibrium and complementarity problems. However, due to the fact that the feasible domain of the MPEC problem does not satisfy most of the constraint specifications, especially the M-F constraint specification, it is not satisfied at any point in the feasible domain, so it is very difficult for this kind of problem to be solved in theory and algorithm. In the past twenty years, great achievements have been made in the research of MPEC problem in theory and algorithm, but there are still many problems to be solved. In this paper, based on projection function and smoothing theory, we study several kinds of mathematical programming problems with equilibrium constraints and multi-objective optimization problems with equilibrium constraints by using combined homotopy algorithm. In this paper, the following results are obtained. 1. The mathematical programming problem with bounded box constrained variational inequalities is studied. First, the variational inequality valence in the problem is transformed into a nonsmooth equation with projection function. Then, based on the characteristics of box constraint set, a parameter equality constraint is constructed by using Cabriel-More smooth function to approximate the nonsmooth part of the equation. The homotopy equation is constructed for the KKT system of the transformed mathematical programming problem. This method does not need to assume that the function F has strong monotonicity, nor does it need to introduce additional variables, and it also facilitates the selection of initial points in subsequent calculations. Finally, the existence of homotopy path and the convergence of a wide range are proved. The feasibility and validity of the algorithm are verified by numerical experiments, and a homotopy algorithm for complementary constrained mathematical programming is established. Firstly, the mathematical programming problem with complementary constraints is transformed into a general nonlinear programming problem with nonsmooth function constraints. Then the nonsmooth equality constraints are transformed into the equality constraints of smooth functions by smoothing method. Thus, the above mathematical programming problem is transformed into a smooth function constrained mathematical programming problem, thus avoiding the introduction of more multiplier variables. The homotopy equation is constructed for the KKT system of the final smooth programming problem. The existence and convergence of the homotopy path are proved. It is also proved that the obtained KKT point is the C-stable point of the original problem. A numerical example is used to verify the feasibility and validity of the algorithm, and a new homotopy algorithm is constructed to solve the multi-objective optimization problem with equilibrium constraints. Firstly, the original problem is equivalent to a general multiobjective problem with KKT system by using SBCQ constraint specification, and the KKT system mentioned above is transformed into a smooth equality constraint, and then a multiobjective programming problem with equality and inequality constraints is obtained. Finally, the homotopy equation is constructed for the KKT system of the transformed equivalent problem, and the existence and convergence of the homotopy path are proved. Finally, the feasibility and validity of the proposed algorithm are proved by numerical experiments. Finally, the equilibrium constrained programming problem defined by variational inequalities on a general closed convex set is discussed. By introducing the concept of infinite solution, the solved MPEC problem is transformed into a single-layer optimization problem with projection function. Finally, by using smoothing method, the final single-layer optimization problem is transformed into an optimization problem with smooth function constraints. The homotopy equation is constructed for its KKT system, and the existence and convergence of homotopy path are proved, and an example is given.
【学位授予单位】：吉林大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O221

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