随机二阶锥互补问题理论与算法研究及其应用

项目名称： 随机二阶锥互补问题理论与算法研究及其应用

项目编号： No.11501275

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 罗美菊

作者单位： 辽宁大学

项目金额： 18万元

中文摘要： 二阶锥规划作为数学规划的重要分支有着重要而又广泛的应用前景。由于二阶锥规划的KKT条件可转化为二阶锥互补问题（SOCCP），因此，SOCCP是比二阶锥规划应用更广泛的均衡优化问题。由于实际问题的多样化，经常会涉及随机因素，漠视这些因素会导致决策失误。本项目将研究应用更广泛的随机二阶锥互补问题 (SSOCCP)。首先，本项目将建立求解SSOCCP的期望值模型、期望残差最小化模型、条件风险价值模型，提出相应模型的样本均值近似问题，并分别开发求解算法，考虑相应的收敛性及优化理论。其次，对于随机变量分布未知的情形，对上述三种模型将通过构造逼近分布集合，分别建立相应的逼近分布鲁棒优化模型，并分别考虑逼近问题最优值及最优解的收敛性。最后，在应用方面，建立求解鲁棒随机纳什均衡问题的SSOCCP模型，给出不同算例，利用提出的确定性模型及相应算法求解，分析数值结果，说明不同模型的优缺点，证实算法的可行性。

中文关键词： 随机二阶锥互补问题；二阶锥互补函数；样本均值近似；收敛性；鲁棒随机纳什均衡

英文摘要： As an important branch of mathematical programming, second-order cone programming has an important and wide-ranging application prospect. Since the KKT conditions of second-order cone programming can be converted into the second-order cone complementarity problems (SOCCP), compared with second-order cone programming, SOCCP is a wide equilibrium optimization problem. As the diversification of practical problems, stochastic factors are usually involved, and a wrong decision will be made if disregarding these factors. This proposal will focus on stochastic second-order cone complementarity problems (SSOCCP) that has a wider application field. Firstly, an expected value model, an expected residual minimization model, a condition value-at-risk model and their sample average approximation problems will be presented for solving SSOCCP. Algorithms, convergence analysis and optimization theory will be considered respectively. Secondly, if the distribution of the stochastic variables can't be obtained, we will present approximation distribution robust optimization problems for the above three models respectively by constructing approximation distribution sets. Moreover, the convergence results of the optimal values and optimal solutions for the approximation problems will be considered. Finally, in applications, we will establish a SSOCCP model for solving robust stochastic Nash equilibrium problems. Different numerical examples will be given and solved by using the presented deterministic models and corresponding algorithms. Through the numerical results, the advantages and disadvantages of different models will be analyzed and the feasibility of the algorithms will be confirmed.

英文关键词： Stochastic second-order cone complementarity problems;Second-order cone complementarity function;Sample average approximation;Convergence;Robust stochastic Nash equilibrium