随机二阶锥互补问题理论与算法研究及其应用

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.

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。

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。本项目研究应用更广泛的随机二阶锥互补问题 (SSOCCP)。但是由于随机因素的存在，SSOCCP通常情况下无解。为此，本项目建立求解SSOCCP的三种确定性模型，即期望值模型、期望残差最小化模型、条件风险价值模型。并将这三种确定性模型的解视为随机二阶锥互补问题的解。由于通常情况下，数学期望都很难计算，本项目应用样本均值近似方法分别给出期望值模型、期望残差最小化模型、条件风险价值模型相应的样本均值近似问题，并在理论上分别考虑相应近似问题的收敛性及优化理论。在应用方面，本项目考虑随机天然气传输问题及径向网络中随机最优潮流问题，并将所提出的求解模型及方法应用到实际问题中。