CAREER: Applications of Convex Programming in Combinatorial Optimization: A Mathematical, Algorithmic and Computational Study

半定规划问题（SDP），包括实现数值稳定 内点算法，可行集的数学性质，以及 特别是SDP在组合优化中的应用， 研究了 首先，原始-对偶内点SDP的两个版本 算法正在实施。一个版本是在MATLAB中，目的是作为一个 易于使用和改变软件的小到中等大小的问题。 另一种实现是在FORTRAN 90中，旨在实现高性能 并行和向量/流水线机器，旨在解决大规模 问题 此外，SDP的活动集方法的可行性正在研究中 研究了SDP在一般0-1整数规划中的应用 问题正在研究中。分支和约束SDP技术的可行性 它在每个阶段解决SDP问题，但可能需要更少的分支 整体正在调查中。 如果发现这种办法切实可行， 将开发通用程序。 在理论研究中， 目前正在研究SDP的以下应用： 图的Lovasz数的计算复杂性， 最小顶点覆盖问题，在一般匹配问题中的应用， 在仿射变换下的SDP的可行集的多面体。的 线性半定规划原对偶方法与 二次约束二次规划问题 研究了此外，正在调查的这些锥体，其相关的 优化问题导致互补关系是双线性形式; 规划，二次约束二次规划 和SDP是这些问题的例子。 对于教学计划，积极 参与帮助RUTCOR建立一个硕士课程的计划。 研究生院的半定规划和计算生物学课程，以及一门旨在调查 正在开发各种可用于优化问题的软件。在 本科层次的课程教学在附属部门的 计算机科学，数学和统计学。此外，正在编写一本关于社会发展方案的教科书。

The semidefinite programming problem (SDP), including implementation of numerically stable interior point algorithms, mathematical properties of the feasible sets, and in particular applications of SDP in combinatorial optimization are being investigated. First, two versions of the primal-dual interior point SDP algorithm are being implemented. One version is in MATLAB and is intended as an easy to use and change software for small to medium size problems. Another implementation is in FORTRAN 90 and is intended for high performance parallel and vector/pipelined machines and is meant to solve large scale problems. In addition feasibility of active set methods for SDP is being investigated. Next applications of SDP in general 0-1 integer programming problems are being studied. The feasibility of branch and bound SDP techniques which solve an SDP problem at each stage but may require far fewer branchings overall is being investigated. If such an approach is found to be practical, general purpose programs will be developed. In theoretical investigations, the following applications of SDP are being studied: parallel computational complexity of the Lovasz number of graphs, approximation algorithms for the minimum vertex cover problem, applications to the general matching problem, and polyhedrality of the feasible sets of SDPs under affine transformations. The connection of primal--dual method of linear and semidefinite programs to the problem of quadratically constrained quadratic programming is being investigated. In addition, those cones are being investiagted whose associated optimization problem leads to complementarity relations that are bilinear forms; programming, quadratically constrained quadratic programming and SDP are examples of such problems. For teaching plans, active participation in helping RUTCOR establish a masters program is planned. Courses in semidefinite programming and in computational biology in graduate school, and a course designed to survey various software available for optimization problems are being developed. At the undergraduate level teaching of courses in the affiliated departments of computer science, mathematics and statistics is expected. Also a textbook on SDP is in preparation.