【作者】 迟晓妮；

【导师】 刘三阳；

【作者基本信息】 西安电子科技大学 ， 应用数学， 2005， 硕士

【摘要】 二次锥规划是在有限个二次锥的笛卡儿乘积的仿射子空间之交上极小化或极大化一个线性函数。其约束是非线性的，但却是凸的，因此二次锥规划是凸规划。二次锥规划包括线性规划和二次约束下的凸二次规划等，却是半定规划的特例。由于其广泛应用及原-对偶内点算法的迅速发展，二次锥规划已经成为数学规划领域的一个重要的研究方向。 本文首先简述了二次锥规划的基本知识，包括二次锥规划的理论、算法和研究现状，然后介绍了在二次锥规划的算法方面所做的一些工作，具体如下： 1．本文给出了二次锥规划的一种原-对偶非精确不可行内点算法。该算法允许搜索方向有相对较大的误差，且不要求迭代点的可行性。在相对不精确的假设下，利用该算法可找到二次锥规划的ε-近似解。 2．在光滑Fischer-Burmeister函数的基础上，本文给出了二次锥规划的一种新的光滑牛顿法。该方法所采用的系统不是等价于中心路径条件，而是等价于最优性条件本身。算法对初始点没有任何限制，且具有Q-二阶收敛速度。更多还原

【Abstract】 The second-order cone programming (SOCP) problem is to minimize or maximize a linear function over the intersection of an affine space with the Cartesian product of a finite number of second-order cones. It is well known that the constraint of the second-order cone programming is nonlinear , but convex, so the second-order cone programming is a convex optimization problem. The second-order cone programming unifies several problems, such as linear programming and quadratically constrained convex quadratic programming, but it is a special case of semidefinite programming. Because of the quick development of its primal-dual interior-point algorithms and its wide applications, the second-order cone programming is an important research field in mathematical programming.In the paper, firstly the theory, algorithm, and recent research of the second-cone programming is summarized, then our some work in algorithms is introduced. For detail, we conclude them as follows:1. A primal-dual inexact infeasible interior-point algorithm for the second-order cone programming is presented in this paper. This algorithm allows the search direction that is calculated with only moderate accuracy, and does not require feasibility of the iteration points. Under a mild assumption on the inexactness, we show that the algorithm can find an e -approximate solution of the second-order cone programming.2. Based on smoothing the Fischer-Burmeister function, a new smoothing Newton method is presented in this paper. The system which is employed in this method is equivalent to the optimality conditions and not to the central path conditions. This algorithm does not have restrictions regarding its starting point and it is Q-quadratically convergent.更多还原