p阶锥规划与互补问题的理论和算法研究

p-order cone programming and complementarity problems are a special kind of asymmetric cone programming and complementarity problems. The problem not only has a broad application background in the practical problems, but also it is a wider class of optimization problem better than second-order cone programming and complementarity problem in terms of mathematical optimization. Therefore, the topic research has important theoretical significance and practical application value. At present, the study on p-order programming and complementarity problem is scarce, and there are very little literature on the study of this project. Hence, the project aim on p-order cone programming and complementarity problems gives further theoretical and algorithmic research. Details are as follows: (1) For p-order cone programming and complementarity problems, we investigate the existence of solution, the boundedness of solution set and the Lipschitz continuity of the solution, etc. (2) For solving p-order cone complementarity problems, we try hard to construct a type of NR merit functions, and study the boundedness of level set of these merit functions, the optimality of the stable point and error bound analysis. (3) For solving p-order cone programming problems, we attempt to establish corresponding NR nonsmooth and smooth algorithm, and give the theoretical analysis about the convergence of the designed algorithms. (4) We give numerical experiments for the design of the algorithms through computation. The results of the study on this project not only enriches the theory knowledge of p-order cone programming and complementarity problems, but also provides a practical and effective calculation methods.

p阶锥规划与互补问题是一类特殊的非对称锥规划与互补问题。在实际问题中不仅有着广泛的应用背景；而且从数学优化方面来看，它也是比二阶锥规划问题更为广泛的一类优化问题。因此，对此课题的研究具有重要的理论意义和实际的应用价值。目前，对p阶锥规划与互补问题的研究比较稀少，只见到少量的文献对此类问题有过研究。于是，本项目旨在对p阶锥规划与互补问题的理论与算法进行进一步的研究。其具体内容为：(1) 探讨p阶锥规划与互补问题解集的非空有界性和解的Lipschitz连续性等；(2) 对于求解p阶锥互补问题，探讨NR型效用函数的水平集有界性、稳定点的最优性以及误差界理论等；(3) 对于求解p阶锥规划问题，建立相应的NR型非光滑和光滑型算法，并对所设计的算法进行收敛性理论分析；(4) 对设计的算法编制程序进行数值试验。该研究成果不仅能够丰富p阶锥规划与互补问题的理知识，而且还提供了一种实用有效的计算方法。

本项目考查研究了欧氏空间中p阶锥规划问题与p阶锥互补问题，研究内容主要体现在一下两部分内容。第一部分内容：考查研究了p阶锥凸规划问题的解集结构特征和p阶锥上的投影等相关内容，主要贡献如下：（1）探讨了p阶锥凸规划问题的解集结构特征；（2）研究了欧氏空间中元素到p阶锥上的投影表达形式和元素的两种谱分解形式及其相关结论；（3）针对p阶锥互补问题，考查了解的Lipschitz连续性的相关结论。第二部分内容的主要贡献体现在对求解p阶锥规划问题和p阶锥问题的数值算法进行了理论和算法分析，主要有：（1） 针对p阶锥互补问题，构造了几类效用函数，并讨论了这些效用函数的水平集有界性以及误差界理论等问题；（2）利用一类锥函数，考查了求解由特殊的p阶锥（即二阶锥）导出的等式与不等式系统及其相关性质；（3）提出了求解二阶锥绝对值方程问题的光滑化型算法，并给出了算法的相关理论分析和数值计算结果。