非线性半定规划的非退化性与强适性内点方法研究

项目名称： 非线性半定规划的非退化性与强适性内点方法研究

项目编号： No.11271107

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 刘新为

作者单位： 河北工业大学

项目金额： 64万元

中文摘要： 半定规划是当前线性和非线性规划领域的一个研究热点，它在自动控制和数字图象处理等领域有着广泛的应用。内点方法已经在求解线性半定规划方面取得了成功，一些求解非线性半定规划的内点方法正在被开发。本项目旨在已有研究工作的基础上，结合半定规划具有半定约束和大规模的特点，发展一类新的更具强适性的非线性半定规划内点方法，建立其相关的全局和局部收敛性理论，并应用于矩阵还原等数字图象恢复问题。在此基础上，研究并提出线性和非线性半定规划新的易于检验的非退化性条件。算法的强适性主要体现在三个方面：一是算法及其理论不要求任何约束非退化性假设；二是对于非线性半定规划，算法的子问题不含有半定约束，因此有利于求解大规模非线性半定规划和改进算法效率；三是它总可以找到半定规划有意义的稳定点。希望通过本项目的研究，有助于进一步揭示半定规划问题的本质（如保证强对偶性条件等），并开发更加有效的求解非线性半定规划和锥规划的算法。

中文关键词： 内点方法；非线性规划；梯度法；分裂方法；半定规划

英文摘要： Semidefinite programming (SDP) has extensive applications in practical fields, such as automatic control, digital image processing, etc., and has attracted great interests of many researchers in linear and nonlinear programming area. Interior-point approach has succeeded in solving SDP, some new interior-point methods for nonlinear SDP have been developed. Based on our completed research works and obtained results, this project is to propose a new kind of nondegeneracy condition which can be examined easily and a class of robust interior-point methods, establish the global and local convergence theories, and apply these methods to problems such as matrix completion arising from the digital image restoration and matrix inequalities from automatical control. Comparing with the existing methods, these robust methods are expected to have the following characteristics: Firstly, their global convergences are not dependent on any constraint nondegeneracy condition such as the strict feasibility of primal and dual problems for linear semidefinite programming; Secondly, for nonlinear SDP, the subproblems do not have any positive semidefinite constraint, thus is suitable for large-scale nonlinear SDP and may be helpful to improve the efficiency of existing methods for SDP; Thirdly, they can always find some points in sens

英文关键词： interior-point method；nonlinear program；gradient method；splitting method；semidefinite program