• 论文 •

### 不等式约束优化一个基于滤子思想的广义梯度投影算法

1. 1. 玉林师范学院数学与信息科学学院, 广西玉林 537000;
2. 广西大学数学与信息科学学院, 南宁 530004;
3. 上海大学理学院, 上海 200444;
4. 广西师范学院数学与信息科学学院, 南宁 530001
• 收稿日期:2012-11-16 出版日期:2013-05-15 发布日期:2013-05-13
• 通讯作者: 简金宝
• 基金资助:

国家自然科学基金(11271086)、广西自然科学基金(2011GXNSFD018022)和广西高校人才小高地建设创新团队资助计划.

Jian Jinbao, Ma Pengfei, Xu Qingjuan. A GENERALIZED GRADIENT PROJECTION METHOD BASED ON THE IDEA OF FILTER FOR INEQUALITY CONSTRAINED OPTIMIZATION[J]. Mathematica Numerica Sinica, 2013, 35(2): 205-214.

### A GENERALIZED GRADIENT PROJECTION METHOD BASED ON THE IDEA OF FILTER FOR INEQUALITY CONSTRAINED OPTIMIZATION

Jian Jinbao1,2, Ma Pengfei3, Xu Qingjuan4

1. 1. School of Mathematics and Informatics Science, Yulin Normal University, Yulin 537000, Guangxi, China;
2. School of Mathematics and Informatics Science, Guangxi University, Nanning 530004, China;
3. Shanghai University College of Science, Shanghai 200444, China;
4. Mathematics and Information Science,Teachers Education University, Nanning 530001, Guangxi, China
• Received:2012-11-16 Online:2013-05-15 Published:2013-05-13

In this paper, optimization problems with nonlinear inequality constraints are discussed. Based on the idea of the filter algorithm, a new generalized gradient projection algorithm is proposed. The proposed method uses neither a penalty function, nor a strict filter. At each iteration of the proposed algorithm, the search direction is yield by just on explicit generalized gradient projection. The step-size is selected such that either the value of the objective function or the measure of the constraint violations is sufficiently reduced by a Armijo line search technique. The main properties of the proposed algorithm as follows: don’t need to assume the boundness of iteration sequence; don’t need any restoration phase which is necessary for filter methods; the scale and the computation cost are further decreased by using the ε-active set. The algorithm is globally convergent under suitable assumptions. Finally, some elementary numerical experiments are reported.

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 [1] Rosen J B. The Gradient Projection Method of Nonlinear Programming, Part I, Linear Constraints[J]. SIAM J Appl Math, 1960, 8(1): 181-21.[2] Rosen J B. The gradient projection method for nonlinear programming part I linear constraints[J].SIAM J Appl Math, 1961, 9(4): 514-532.[3] 堵丁柱, 孙捷. 一个新的梯度投影方法[J]. 计算数学, 1983, 4: 378-386.[4] 章祥荪. 关于梯度投影法的收敛性证明[J]. 应用数学学报,1985, 1(1): 125-128.[5] Jian Jinbao. A new feasible descent algorithm combining SQP with generalized projection foroptimization problems without strict complementarity[J]. Appl Math Comput, 2005, 162(3):1065-1081.[6] 赖炎连, 简金宝.初始点任意的一个非线性优化的广义梯度投影法[J]. 系统科学与数学, 1995,15(4): 374-380.[7] Jian Jinbao, Ma Guodong, Guo Chuanhao. A new ε-generalized projection method of stronglysub-feasible directions for inequality constrained optimization[J]. Journal of System Science andComplex, 2011, 24(1): 604–618.[8] Fletcher R, Leyffer S. Nonlinear programming without a penalty function[J]. Math Program, 2002,91: 239-269.[9] Fletcher R, Gould N, Leyffer S, et al. Global convergence of a trust-region SQP-filter algorithmfor general nonlinear programming[J]. SIAM J Optim, 2002, 13(3): 635-659.[10] Ulbrich S. On the superlinear local convergence of a filter-SQP methods[J]. Math Program, 2004,100: 217-245.[11] Wäechter A, Biegler. Line search filter methods for nonlinear programming : motivation andglobal convergence[J]. SIAM J Optim, 2005, 16(1): 1-31.[12] Wäechter A, Biegler. Line search filter methods for nonlinear programming: local convergence[J].SIAM J Optim, 2005, 16(1): 32-48.[13] Nie Puyan. Sequential penalty quadratic programming filter methods for nonlinear programming[J]. Nonlinear Anal-Real, 2007, 8(1): 118-129.[14] Li Duoquan. A new sqp-filter method for solving nonlinear programming problems[J]. Journal ofcomputational mathematics, 2006, 24(5): 609-634.[15] 简金宝,光滑约束优化快速算法——理论分析与数值实验[M]. 北京, 科学出版社,2010.[16] Ulbrich, Ulbrich M. Nonmonotone trust region methods for nonlinear equality constrained optimizationwithout a penalty function[J]. Mathematical Programming, Series B, 95(1), 103-105,2003.[17] Gould N, Toint Ph L. Nonlinear programming without a penalty function or a filter[J]. MathProgram, 2010, 122(1): 155-196.[18] Gould N, Robinson D P, Toint Ph L. Corrigendum: Nonlinear programming without a penaltyfunction or a filter, Technical Report, Namur Center for Complex Systems, 2011.[19] Liu Xinwei, Yuan Yaxiang. A sequential quadratic programming method without a penalty functionor a filter for nonlinear equality constrained optimization[J]. SIAM J Optim, 2011, 21(2):545-571.[20] Solodov M V. Global convergence of an SQP method without boundedness assumptions on anyof iterative sequences[J]. Math Program, 2009, 118: 1-12.[21] Zhou Guanglu. A modified SQP method and its global convergence[J]. Journal of Global Optimization,1997, 11(2): 193-205.[22] Hock W, Schittkowski K. Test examples for nonlinear Programming Codes. Lecture Notes inEconomics and Mathematical Systems[J]. Springer, 1981, 187.
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