一类极大极小随机规划逼近问题最优解集的上半收敛性

本文首先将极大极小随机规划等价的转化为一个二层随机规划,在下层初始随机规划最优解集为多点集的情形下,给出下层随机规划逼近问题最优解集集值映射关于上层决策变量参数的上半收敛性和最优值函数的连续性.然后将上层随机规划等价转化为以上层和下层决策变量作为整体决策变量,以下层规划最优解集的图作为约束条件的单层规划,并在下层初始随机规划最优解集的图为正则的条件下,得到上层随机规划逼近问题最优解集关于最小信息概率度量收敛的上半收敛性.     

二层随机规划逼近问题最优解集的上半收敛性

在下层初始随机规划问题可行解集上引入了正则的概念,并在下层初始随机规划最优解唯一的条件下,利用上图收敛理论,给出了下层随机规划逼近问题的任意一个最优解向量函数都连续收敛到下层初始随机规划问题的唯一最优解向量函数.然后将下层随机规划的最优解向量函数反馈到上层随机规划的目标函数和约束条件中,得到了上层随机规划逼近问题的最优解集关于最小信息概率度量收敛的上半收敛性.     

二层随机规划逼近最优解集的上半收敛性

下层随机规划以上层决策变量作为参数,而上层随机规划是以下层随机规划的唯一最优解作为响应的一类二层随机规划问题,首先在下层随机规划的原问题有唯一最优解的假设下,讨论了下层随机规划的任意一个逼近最优解序列都收敛于原问题的唯一最优解,然后将下层随机规划的唯一最优解反馈到上层,得到了上层随机规划逼近最优解集序列的上半收敛性.     

一类随机规划逼近最优值和最优解集的收敛性

本文提出强上图收敛的概念,讨论了逼近随机规划的目标函数序列的强上图收敛性,研究了逼近随机规划最优值和最优解集的收敛性条件,得到了一类随机规划逼近最优值和最优解集的收敛性.     

随机规划ε-逼近最优解集的Hausdorff收敛性

本文研究了随机规划ε-逼近最优解集的Haudorff收敛性条件,证明了随机规划逼近最优值的收敛性,并利用此结果给出了随机规划ε-逼近最优解集Haudorff收敛的一个充分条件.     

一类概率约束规划逼近最优解集的上半收敛性

对一类概率约束规划逼近最优解集的上半收敛性进行了研究.利用概率测度弱收敛的特征,给出了概率约束规划可行集的收敛性条件,得到了概率约束规划逼近最优解集的上半收敛性.     

概率约束规划逼近最优解集的上半收敛性

本文讨论了概率约束规划目标函数的连续收敛性,并利用概率测度弱收敛的特征给出了概率约束规划可行集的收敛性条件,得到了概率约束规划逼近最优解集的上半收敛性.     

不确定规划逼近问题最优解的几乎处处上半收敛性

对不确定规划经验逼近问题的最优解的几乎处处上半收敛性进行了研究。首先将带有约束的不确定规划问题转化成与其等价的无约束的不确定优化问题,然后将经验测度替代不确定测度得到不确定规划的经验逼近模型,并得出逼近问题的目标函数序列的几乎处处上图收敛性,最后利用上图收敛性理论,给出了不确定规划经验逼近最优解集的几乎处处上半收敛性。     

随机规划经验逼近问题最优解集的几乎处处下半收敛性

本文给出了随机规划经验逼近最优解集几乎处处下半收敛的一个充分条件,并由此得到随机规划经验逼近最优解集几乎处处Hausdorff收敛的一个充分条件.     

非线性参数规划问题ε-最优解集集值映射的连续性

对非线性参数规划问题ε-最优解集集值映射的连续性条件进行了研究.首先在可行集集值映射局部有界且正则的条件下,讨论了非线性参数规划问题最优值函数的连续性,然后针对ε-最优解集集值映射的结构特征并利用此结果和集值分析理论,给出了非线性参数规划问题ε-最优解集集值映射连续的一个充分条件.     

On distributionally robust optimization problems with k-th order stochastic dominance constraints induced by full random quadratic recourse

In this paper, we consider the optimization problems with k-th order stochastic dominance constraint on the objective function of the two-stage stochastic programs with full random quadratic recourse. By establishing the Lipschitz continuity of the feasible set mapping under some pseudo-metric, we show the Lipschitz continuity of the optimal value function and the upper semicontinuity of the optimal solution mapping of the problem. Furthermore, by the Hölder continuity of parameterized ambiguity set under the pseudo-metric, we demonstrate the quantitative stability results of the feasible set mapping, the optimal value function and the optimal solution mapping of the corresponding distributionally robust problem.     

Bilevel problems over polyhedra with extreme point optimal solutions

Bilevel programming involves two optimization problems where the constraint region of the upper level problem is implicitly determined by another optimization problem. In this paper we focus on bilevel problems over polyhedra with upper level constraints involving lower level variables. On the one hand, under the uniqueness of the optimal solution of the lower level problem, we prove that the fact that the objective functions of both levels are quasiconcave characterizes the property of the existence of an extreme point of the polyhedron defined by the whole set of constraints which is an optimal solution of the bilevel problem. An example is used to show that this property is in general violated if the optimal solution of the lower level problem is not unique. On the other hand, if the lower level objective function is not quasiconcave but convex quadratic, assuming the optimistic approach we prove that the optimal solution is attained at an extreme point of an ??enlarged?? polyhedron.     

Quantitative stability of full random two-stage stochastic programs with recourse

In this paper, we consider quantitative stability analysis for two-stage stochastic linear programs when recourse costs, the technology matrix, the recourse matrix and the right-hand side vector are all random. For this purpose, we first investigate continuity properties of parametric linear programs. After deriving an explicit expression for the upper bound of its feasible solutions, we establish locally Lipschitz continuity of the feasible solution sets of parametric linear programs. These results are then applied to prove continuity of the generalized objective function derived from the full random second-stage recourse problem, from which we derive new forms of quantitative stability results of the optimal value function and the optimal solution set with respect to the Fortet–Mourier probability metric. The obtained results are finally applied to establish asymptotic behavior of an empirical approximation algorithm for full random two-stage stochastic programs.     

Continuity and Stability of a Quadratic Mixed-Integer Stochastic Program

For the two-stage quadratic stochastic program where the second-stage problem is a general mixed-integer quadratic program with a random linear term in the objective function and random right-hand sides in constraints, we study continuity properties of the second-stage optimal value as a function of both the first-stage policy and the random parameter vector. We also present sufficient conditions for lower or upper semicontinuity, continuity, and Lipschitz continuity of the second-stage problem's optimal value function and the upper semicontinuity of the optimal solution set mapping with respect to the first-stage variables and/or the random parameter vector. These results then enable us to establish conclusions on the stability of optimal value and optimal solutions when the underlying probability distribution is perturbed with respect to the weak convergence of probability measures.     

Lipschitz continuity of the optimal value function and KKT solution set in indefinite quadratic programs

When all the involved data in indefinite quadratic programs change simultaneously,we show the locally Lipschtiz continuity of the KKT set of the quadratic programming problem firstly, then we establish the locally Lipschtiz continuity of the KKT solution set. Finally, the similar conclusion for the corresponding optimal value function is obtained.     

Quantitative stability of full random two-stage problems with quadratic recourse

ABSTRACT

In this paper, we discuss quantitative stability of two-stage stochastic programs with quadratic recourse where all parameters in the second-stage problem are random. By establishing the Lipschitz continuity of the feasible set mapping of the restricted Wolfe dual of the second-stage quadratic programming in terms of the Hausdorff distance, we prove the local Lipschitz continuity of the integrand of the objective function of the two-stage stochastic programming problem and then establish quantitative stability results of the optimal values and the optimal solution sets when the underlying probability distribution varies under the Fortet–Mourier metric. Finally, the obtained results are applied to study the asymptotic behaviour of the empirical approximation of the model.     

An optimal method for stochastic composite optimization

This paper considers an important class of convex programming (CP) problems, namely, the stochastic composite optimization (SCO), whose objective function is given by the summation of general nonsmooth and smooth stochastic components. Since SCO covers non-smooth, smooth and stochastic CP as certain special cases, a valid lower bound on the rate of convergence for solving these problems is known from the classic complexity theory of convex programming. Note however that the optimization algorithms that can achieve this lower bound had never been developed. In this paper, we show that the simple mirror-descent stochastic approximation method exhibits the best-known rate of convergence for solving these problems. Our major contribution is to introduce the accelerated stochastic approximation (AC-SA) algorithm based on Nesterov’s optimal method for smooth CP (Nesterov in Doklady AN SSSR 269:543–547, 1983; Nesterov in Math Program 103:127–152, 2005), and show that the AC-SA algorithm can achieve the aforementioned lower bound on the rate of convergence for SCO. To the best of our knowledge, it is also the first universally optimal algorithm in the literature for solving non-smooth, smooth and stochastic CP problems. We illustrate the significant advantages of the AC-SA algorithm over existing methods in the context of solving a special but broad class of stochastic programming problems.     

Continuity of the optimal value function in indefinite quadratic programming

This paper characterizes the continuity property of the optimal value function in a general parametric quadratic programming problem with linear constraints. The lower semicontinuity and upper semicontinuity properties of the optimal value function are studied as well.     

Solving discrete linear bilevel optimization problems using the optimal value reformulation

In this article, we consider two classes of discrete bilevel optimization problems which have the peculiarity that the lower level variables do not affect the upper level constraints. In the first case, the objective functions are linear and the variables are discrete at both levels, and in the second case only the lower level variables are discrete and the objective function of the lower level is linear while the one of the upper level can be nonlinear. Algorithms for computing global optimal solutions using Branch and Cut and approximation of the optimal value function of the lower level are suggested. Their convergence is shown and we illustrate each algorithm via an example.     

Quantitative stability of mixed-integer two-stage quadratic stochastic programs

For our introduced mixed-integer quadratic stochastic program with fixed recourse matrices, random recourse costs, technology matrix and right-hand sides, we study quantitative stability properties of its optimal value function and optimal solution set when the underlying probability distribution is perturbed with respect to an appropriate probability metric. To this end, we first establish various Lipschitz continuity results about the value function and optimal solutions of mixed-integer parametric quadratic programs with parameters in the linear part of the objective function and in the right-hand sides of linear constraints. The obtained results extend earlier results about quantitative stability properties of stochastic integer programming and stability results for mixed-integer parametric quadratic programs.