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是半正定的本质约束下被最小化。一些问题可以归结为SCP问题，包括线性规划问题、具有凸二次不等式约束的凸二次问题、矩阵范数最小化问题以及各种最大和最小特征值问题。另外，SDP在工程、组合优化、统计等领域有着广泛的应用，目前已有多种线性规划的内点算法可以推广到SDP问题中。就像在线性规划中一样，这些方法中的许多都是多项式收敛的，并且在实践中执行得非常有效。特别是，原-对偶内点方法及其高阶变种是求解SDP问题的一种非常有效的方法。与线性规划相比，SDP的原始-对偶算法中使用的牛顿搜索方向有很多种计算方法。因此，SDP的原-对偶方法的理论比LP的理论要困难得多。本项目致力于SDP问题的理论和算法实现的发展。本研究的目的包括：1)增进关于SDP的多项式和超线性收敛分析理论的知识；2)研究SDP连续轨迹的存在性和渐近性；3)发展超线性收敛的高阶原始-对偶方法来求解没有严格互补解的SDP问题；4)发展内点原始-对偶算法来解决更一般的锥规划问题；5)基于非线性规划技术发展解决特殊结构SDP问题的新方法；6)实现这些新方法并将它们与现有的SDP方法进行比较；7)将原始-对偶内点算法扩展到非线性SDP和互补问题的背景下，这项研究将导致新的或改进的算法来寻找大规模优化问题的精确或近似解，这些问题在工业、金融、科学和工程中的应用非常广泛。

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.