一类带概率互补约束的随机优化问题的最优性条件

doi:10.15960/j.cnki.issn.1007-6093.2017.02.003

运筹学学报

一类带概率互补约束的随机优化问题的最优性条件

陈林¹ 杨新民^2,*

1. 四川大学数学学院, 成都 610064; 2. 重庆师范大学数学科学学院, 重庆 400047

收稿日期:2017-04-10 出版日期:2017-06-15 发布日期:2017-06-15
通讯作者: 杨新民 E-mail: xmyang@cqnu.edu.cn
基金资助:
国家自然科学基金(No. 11431004)

Optimality conditions for a class of stochastic optimization problems with probabilistic complementarity constraints

CHEN Lin¹YANG Xinmin^2,*

1. Collage of Mathematics, Sichuan University, Chengdu 610064, China; 2. School of Mathematical Sciences, Chongqing Normal University, Chongqing 400047, China

Received:2017-04-10 Online:2017-06-15 Published:2017-06-15

摘要/Abstract

摘要：

主要讨论了一类带概率互补约束的随机优化问题的最优性条件. 首先利用一类非线性互补(NCP)函数将概率互补约束转化成为一个通常的概率约束. 然后, 利用概率约束的相关理论结果, 将其等价地转化成一个带不等式约束的优化问题. 最后给出了这类问题的弱驻点和最优解的最优性条件.

关键词: 概率互补约束, NCP 函数, alpha-凹函数, 最优性条件

Abstract:

In this paper, we focus on the optimality conditions for a class of stochastic optimization problem with probabilistic complementarity constraints. By using a kind of nonlinear complementarity (NCP) function, we transform the probabilistic complementary constraint into a chance constraint. By using the theories in chance constraint, we obtain an optimization problem with inequality constraint and then, optimality conditions for weak stationary points and the optimal solutions are given.

Key words: probabilistic complementarity constraints, NCP function, alpha-concave function, optimality conditions

陈林, 杨新民. 一类带概率互补约束的随机优化问题的最优性条件[J]. 运筹学学报, doi: 10.15960/j.cnki.issn.1007-6093.2017.02.003.

CHEN Lin, YANG Xinmin. Optimality conditions for a class of stochastic optimization problems with probabilistic complementarity constraints[J]. Operations Research Transactions, doi: 10.15960/j.cnki.issn.1007-6093.2017.02.003.

参考文献

[1] Charnes A, Cooper W W, Symonds G H. Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil [J]. Management Science, 1958, 4: 235-263.

[2] Miller L B, Wagner H. Chance-constrained programming with joint constraints [J]. Operations Research, 1965, 13: 930-945.

[3] Pr\'{e}kopa A. On probabilistic constrained programming [C]//Proceedings of the Princeton Symposium on Mathematical Programming, Princeton: Princeton University Press, 1970.

[4] Shapiro A, Dentcheva D, Ruszczy\'{n}ski A. Lectures on Stochastic Programming: Modeling and Theory [M]. Philadelphia: SIAM, 2009.

[5] Luedtke J, Ahmed S, Nemhauser G. An integer programming approach for linear programs with probabilistic constraints [J]. Mathematical Programming, 2010, 2: 247-272.

[6] Chen W, Sim M, Sun J, et al. From CVaR to uncertainty set:Implications in joint chance-constrained optimization [J]. Operations Research, 2010, 58: 470-485.

[7] Hong L J, Yang Y, Zhang L W. Sequential convex approximations to joint chance constrained programs: A Monte Carlo approach [J]. Operations Research, 2011, 59: 617-630.

[8] Luo Z Q, Pang J S, Ralph D. Mathematical Programs with Equilibrium Constraints [M]. Cambridge: Cambridge University Press, 1996.

[9] Patriksson M, Wynter L. Stochastic mathematical programs with equilibrium constraints [J]. Operations Research Letters, 1999, 25: 159-167.

[10] Lin G H, Chen X, Fukushima M. Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization [J]. Mathematical Programming, 2009, 116: 343-368.

[11] Xu H F. An implicit programming approach for a class of stochastic mathematical programs with equilibrium constraints [J]. SIAM Journal on Optimization, 2006, 16: 670-696.

[12] Shapiro A. Stochastic mathematical programs with equilibrium constraints [J]. Journal of Optimization Theory and Applications, 2006, 128: 223-243.

[13] Birbil S I, G\"{u}rkan G, Listes O. Solving stochastic mathematical programs with complementarity constraints using simulation [J]. Mathematics of Operations Research, 2006, 31: 739-760.

[14] Meng F, Xu H. Exponential convergence of sample average approximation methods for a class of stochastic mathematical programs with complementarity constraints [J]. Journal of Computational Mathematics, 2006, 24: 733-748.

[15] Scholtes S. Convergence properties of a regularization scheme for mathematical programs with complementarity constraints [J]. SIAM Journal on Optimization, 2001, 11: 918-936.

[16] Zhang J, Zhang Y Q, Zhang L W. A sample average approximation regularization method for a stochastic mathematical program with general vertical complementarity constraints [J]. Journal of Computational and Applied Mathematics, 2015, 28: 202-216.

[17] Facchinei F, Pang J S. Finite-Dimensional Variational Inequalities and Complementary Problems [M]. New York: Springer-Verlag, 2003.

[18] Chen X J, Fukushima M. Expected residual minimization method for stochastic linear complementarity problems [J]. Mathematics of Operations Research, 2005, 30: 1022-1038.

[19] Mangasarian O L, Solodov M V. Nonlinear complementarity as unconstrained and constrained minimization [J]. Mathematical Programming, 1993, 62: 277-297.

[20] Clarke F H. Optimization and Nonsmooth Analysis [M]. New York: John Wiley, 1983.

[21] Xu H F, Zhang D L. Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications [J]. Mathematical Programming, 2009, 119: 371-401.

一类带概率互补约束的随机优化问题的最优性条件

Optimality conditions for a class of stochastic optimization problems with probabilistic complementarity constraints

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 7

编辑推荐

Metrics

本文评价

[1]	陈瑞婷, 徐智会, 高英. 拟凸多目标优化问题近似解的最优性条件[J]. 运筹学学报, 2019, 23(1): 35-44.
[2]	张亮,王燕,李国权. 双值约束非凸三次规化问题的全局最优性条件[J]. 运筹学学报, 2015, 19(2): 83-90.
[3]	李丽花, 高岩, 王隔霞. 一类变结构动态系统的非光滑最优性条件[J]. 运筹学学报, 2013, 17(3): 65-72.
[4]	黎健玲, 谢琴, 简金宝. 均衡约束数学规划的约束规格和最优性条件综述[J]. 运筹学学报, 2013, 17(3): 73-85.
[5]	晁绵涛, 简金宝, 梁东颖. 次b凸函数和次b凸规划[J]. 运筹学学报, 2012, 16(2): 1-8.
[6]	白乙拉, 吕巍. 一类非光滑分布参数系统的可辨识性及最优性条件[J]. 运筹学学报, 2011, 15(2): 119-126.
[7]	张丽丽, 李建宇, 李兴斯. 全局优化问题的一类积分型最优性条件 (英)[J]. 运筹学学报, 2011, 15(1): 1-10.