【作者】 李静；

【导师】 周树民；

【作者基本信息】 武汉理工大学 ， 应用数学， 2005， 硕士

【摘要】 锥规划(conic optimization,简称CO)是一种特殊的凸规划,是线性规划的推广。它指的是在一个仿射空间与一个正则锥的交集上,求线性目标函数的极小或极大值。这个问题总括了线性规划(linear programming,简称LP)、凸二次约束规划(convex quadratic programming,简称QCOP)、半定规划(semidefinite programming,简称SDP)、二次锥规划(second-order conic optimization,简称SOCP)。从它的模型可以看出,它的约束条件和线性规划相比,既是非线性的也是凸约束。近年来,由于它的理论和算法有很大的进展,并且在投资组合优化、最小风险套利、协方差矩阵的逼近等方面得到了广泛的应用,因此成为数学规划领域中一个非常活跃的研究方向。 本文围绕锥规划问题,对锥及其对偶锥的性质进行了研究,解决了一些特殊锥(钝锥、直角锥、优劣钝锥)及其对偶锥之间的关系,并对它们存在的充要条件给予了详细的证明。在此基础上,通过与线性规划作对比,将对偶定理(弱对偶性、强对偶性)、互补松弛定理等推广到锥规划问题中,得到了一些有意义的结论,并且得到了这两个规划的零对偶间隙的存在条件。本文主要由理论研究和应用实践两部分组成。第一部分是理论研究:在线性规划的基础上重点介绍了一种特殊的凸规划类型——锥规划及其对偶锥规划,并介绍了锥规划及其对偶锥规划的发展及其性质。第二部分是应用实践:将所提出的锥规划及其对偶锥规划应用在各个领域,例如最小二乘问题、多项式求解、协方差矩阵估计,以及在其它方面的运用,这些应用无不显示出研究锥规划的必要。本文的具体研究内容如下安排: 第一章介绍了国内外对锥规划及其对偶锥规划的研究现状、模型,指出本文研究要解决的关键问题及研究内容。 第二章介绍了本文要用到的锥及其对偶锥的主要性质,以及几类特殊锥及其对偶锥的性质、充要条件等关键性问题都进行了详细的分析和证明。 第三章是文章的主体部分,主要介绍了锥规划及其对偶锥规划的若干性质。研究主要有:利用Nesterov和Todd的齐次模型判断锥规划与其对偶锥规划解的存在性;类似于线性规划推导出锥规划的KKT-条件;从锥规划的泛对偶性得到锥规划与对偶锥规划的零对偶间隙存在的条件,从而了解泛对偶性与原—对偶更多还原

【Abstract】 Conic optimization (CO) is a particular case of convex programming, and it is also an extension of linear programming.In conic optimization one minimizes (or maximizes) a linear objective function over the intersection of an affine space with a regular cone in finite dimension.This problem comprises of linear programming(LP), convex quadratic programming(QCQP),semidefinite programming(SDP), second-order conic optimization(SOCP).From its model,similar to the linear programming,it is known that the constraint qualification is not only nonlinearly but also convex. In recent years, conic optimization has been one of the most active research areas in mathematical programming because its theory and algotithms have developed greatly and its numerous applications have been found in portfolio optimization, minimum risk arbitrage, and approximating covariance matrices,and so on.On conic optimization, we work over some properties of cone and its dual cone, and solve the relations between special cones (obtuse cone, orthogonal cone, superior and inferior obtuse cones) and their dual cones.Furthermore, sufficient and necessary condition under which they exist are proved detailed . Based on much knowledge, contrasting to linear programming; we extend duality theorem (including weak duality theorem and strong duality theorem), complementary slack theorem to conic optimization. Hence we find out some significative conclusions and existing conditions under which their duality gap is zero of two optimizations.This article is mostly made up of theory study and application practice. In first part it pays attention to theory study. Afterwards, based on linear programming, we introduce some properties of conic optimization and its dual problem, which are a kind of especial convex programming. In second part, we apply these theories to practice, such as least-squares problems, polynomial solution and approximating covariance matrices, and other aspects .All of these applications show that study on conic optimization is necessary. For detail, we conclude them as follows:The models, foreign and internal study situations on conic optimization and its dual problem are introduced in Chapter one, and we point out the key problem and research matter.Chapter two introduces some primary properties of a cone and its dual cone. Furthermore, some properties, sufficient and necessary condition of special cones and their dual cones are analyzed and proved detailed.Chapter three is the most principal part. In third chapter, some important properties of conic optimization and its dual problem are detailed introduced. We utilize Nesterov and Todd’s homogeneous model to estimate the existence of solutions between conic optimization and its dual conic optimization. Similar to linear programming, we give the "KKT" condition of the two optimizations. Through the universal duality in conic convex optimization, we gain the existence condition under which the duality gap of the two programmings is zero. Accordingly, we know the connection of the universal duality and boundness of a primal-dual pair feasible set.We apply these theories to practical applications in chapter four.In chapter five we make many conclusions and bring forward research expectation for the future.conic optimization; dual conic optimization; duality gap; universal duality更多还原