Interior Point Methods: Semidefinite and Nonlinear 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SDP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。此外，SDP还在工程、组合优化和统计等领域有着广泛的应用。众所周知，线性规划的几种内点算法可以推广到SDP问题。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最有效的内点方法。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。本项目致力于解决SDP问题的原始-对偶内点算法的理论和实现的发展。本研究的目标包括：1)发展一种新的SDP的原-对偶内点算法；2)推广SDP的多项式和超线性收敛分析的理论知识；3)研究SDP连续轨迹的存在性和渐近性；4)发展新的高阶SDP的原-对偶方法；5)实现新的算法并将其与现有的原-对偶内点算法进行比较；6)将原-对偶内点算法推广到非线性SDP和互补问题的背景下。

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 SDP 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 enginnering, combinatorial optimization and statistics. 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 the most effective interior point methods. 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 primal-dual interior point algorithms for SDP problems. The objectives of this research project include: 1) developing a new primal-dual interior point algorithm for SDP; 2) advancing the knowledge of the theory of polynomial and superlinear convergence analysis of primal-dual methods for SDP; 3) studying the existence and asymptotic behavior of continuous trajectories for SDP; 4) developing new higher-order primal-dual methods for SDP; 5) implementing the new proposed algorithm and comparing it against existing primal-dual interior point methods; and 6) extending primal-dual interior point algorithms to the context of nonlinear SDP and complementarity problems.