Theory and Implementation of Algorithms for Semi-Definite and Cone Programming

在半定规划（SDP）问题中，对称矩阵变量X的线性函数在X上的线性等式约束和X是半正定的基本约束下被最小化. 一些问题可以转化为SCP问题，包括线性规划，凸二次不等式约束的凸二次问题，矩阵范数最小化以及各种最大和最小特征值问题。 另外，SDP在工程、组合优化和统计学中也有许多应用，目前已知线性规划的几种邻域点算法都可以推广到SDP问题。 在线性规划中，这些方法中的许多是多项式收敛的，并且在实践中非常有效。 特别是原始-对偶邻近点方法及其高阶变式是求解SDP问题的非常有效的方法。 与线性规划相比，有许多方法可以计算SDP的原始-对偶算法中使用的牛顿搜索方向。 由于这个原因，SDP的原始-对偶方法的理论实质上比LP更困难。 本计画致力于发展SDP问题的理论与演算法。 本论文的主要目的是：1）对SDP问题的多项式理论和原-对偶方法的超线性收敛性进行深入的研究，2）研究SDP问题的连续轨线的存在性和渐近性，3）对不具有严格互补解的SDP问题发展超线性收敛的高阶原-对偶方法，4）研究非严格互补解的SDP问题。4）发展了求解更一般的锥规划问题的邻域点原-对偶算法; 5）发展了基于非线性规划技术的求解特殊结构的SDP问题的新方法; 6）实现了这些新方法并与现有的SDP方法进行了比较; 7）将原-对偶内点算法扩展到非线性SDP和互补问题的背景下，这一研究将导致新的或改进的算法，以找到工业、金融、科学和工程中各种应用中出现的大规模优化问题的精确或近似解。

In a semidefinite programming (SDP) problem, a linear function of a symmetric matrix variable X is minimized subject to linear equality constraints on X and the essential constraint that X be positive semidefinite. Several problems can be cast as SCP problems including linear programs, convex quadratic problems with convex quadratic inequality constraints, matrix norm minimization and a variety of maximum and minimum eigenvalue problems. In addition, SDP has many applications in engineering, combinatorial optimization and statistics.Today, it is known that several interior-point algorithms for linear programs can be extended to SDP problems. As in linear programming, many of these methods are polynomially convergent and perform very efficiently in practice. In particular, the class of primal-dual interior-point methods and their higher-order variants are very effective methods for solving SDP problems. In contrast to linear programming, there are many ways one can compute the Newton search directions used in primal-dual algorithms for SDP. For this reason, the theory of primal-dual methods for SDP is substantially more difficult than that for LP. This project addresses the development of the theory and implementation of algorithms for SDP problems. The objectives of this research consist of: 1) advancing the knowledge of the theory of polynomial and superlinear convergence analysis of primal-dual methods for SDP; 2) studying the existence and asymptotic behavior of continuous trajectories for SDP; 3) developing superlinearly convergent higher-order primal-dual methods for SDP problems which do not have strictly complementary solutions; 4) developing interior-point primal-dual algorithms to solve more general classes of cone programming problems; 5) developing new methods for solving specially structured SDP problems based on nonlinear programming techniques; 6) implementing these new approaches and comparing them against existing SDP methods; 7) extending primal-dual interior point algorithms to the context of nonlinear SDP and complementary problems.This research will lead to new or improved algorithms to find exact or approximate solutions to large-scale optimization problems arising in diverse applications in industry, finance, science, and engineering.