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的线性等式约束和X是半正定的基本约束下被最小化。线性规划问题、带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极大极小特征值问题等都可以转化为SDP问题。此外，SDP在工程、组合优化和统计等方面也有广泛的应用。已知几种线性规划的内点算法可以推广到SDP问题。与线性规划一样，这些方法中的许多都是多项式收敛的，在实践中表现得非常有效。其中，一类原对偶内点法及其高阶变异体是最有效的内点法。与线性规划相反，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。因此，SDP的原始对偶方法的理论要比LP的理论困难得多。本项目讨论了SDP问题的原对偶内点算法的理论和实现的发展。本课题的研究目标包括：1)开发一种新的SDP的原对偶内点算法；2)提高了SDP的多项式理论和原始-对偶方法的超线性收敛分析知识；3)研究了SDP连续轨迹的存在性和渐近性；4)发展新的SDP高阶原对偶方法；5)实现新算法，并与现有的原对偶内点方法进行比较；6)将原对偶内点算法推广到非线性SDP和互补问题。

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.