﻿ 二阶锥规划的光滑非精确牛顿法 Smoothing Inexact Newton Method for the Second Order Cone Programming

Vol.04 No.03(2015), Article ID:15898,6 pages
10.12677/AAM.2015.43033

Smoothing Inexact Newton Method for the Second Order Cone Programming

Li Dong1, Siqi Xu2, Jingen Yang1

1College of Mathematics and Information Science, Xinyang Normal University, Xinyang Henan

2Department of Civil Engineering, Xinyang Normal University, Xinyang Henan

Email: dongli9261128@163.com

Received: Jul. 30th, 2015; accepted: Aug. 12th, 2015; published: Aug. 18th, 2015

ABSTRACT

A new smoothing inexact Newton method is presented for solving the second-order cone programming. At each iteration, the method uses an inexact Newton method to solve the system of equations, which saves computation work of smoothing Newton methods. Under weak assumptions, our method is proved to have global and local quadratic convergence. Numerical experiments indicate that the proposed method is quite effective.

Keywords:Second-Order Cone Programming, Smoothing Inexact Newton Method, Convergence

1信阳师范学院数学与信息科学学院，河南 信阳

2信阳师范学院土木工程学院，河南 信阳

Email: dongli9261128@163.com

1. 引言

(P)

(D)

2. 预备知识

(1)

3. 算法描述

(2)

(3)

。利用定义函数如下：

(4)

(5)

(6)

(7)

(8)

(i)上全局Lipschitz连续且处处强半光滑，并且在任意的处连续可微，其雅可比矩阵为

(ii) 如果矩阵行满秩，则对任意的可逆。

(9)

(10)

。由(5)和(7)知，故知

,

4. 算法的收敛性质

,

5. 数值试验

Smoothing Inexact Newton Method for the Second Order Cone Programming[J]. 应用数学进展, 2015, 04(03): 271-276. http://dx.doi.org/10.12677/AAM.2015.43033

1. 1. Alizadeh, F. and Goldfarb, D. (2003) Second-order cone optimization. Mathematical Programming, 95, 3-51. http://dx.doi.org/10.1007/s10107-002-0339-5

2. 2. Chi, X.N. and Liu, S.Y. (2009) A non-interior continuation method for second-order cone optimization. Optimization, 58, 965-979. http://dx.doi.org/10.1080/02331930701763421

3. 3. Chi, X.N. and Liu, S.Y. (2009) A one-step smoothing Newton method for second-order cone programming. Journal of Computational and Applied Mathematics, 223, 114-123. http://dx.doi.org/10.1016/j.cam.2007.12.023

4. 4. Fang, L., He, G.P. and Hu, Y.H. (2009) A new smoothing Newton-type method for second-order cone programming problems. Applied Mathematics and Computation, 215, 1020-1029. http://dx.doi.org/10.1016/j.amc.2009.06.029

5. 5. Tang, J.Y., He, G.P., Dong, L. and Fang, L. (2011) A smoothing Newton method for second-order cone optimizationbased on a new smoothing function. Applied Mathe-matics and Computation, 218, 1317-1329. http://dx.doi.org/10.1016/j.amc.2011.06.015

6. 6. 汤京永, 贺国平 (2012) 一个新的求解二阶锥规划的非内部连续化算法. 应用数学, 1, 26-31.

7. 7. Fukushima, M., Luo, Z.Q. and Tseng, P. (2002) Smoothing functions for second-order-cone complementarity problems. SIAM Journal on Optimization, 12, 436-460. http://dx.doi.org/10.1137/S1052623400380365