一种新的广义次梯度及其性质

数学物理学报 ›› 2015, Vol. 35 ›› Issue (5): 927-935.

一种新的广义次梯度及其性质

黄正刚

重庆理工大学数学与统计学院重庆 400054

收稿日期:2014-04-20 修回日期:2015-03-04 出版日期:2015-10-25 发布日期:2015-10-25
作者简介:黄正刚,hzg@cqut.edu.cn

A New Notion of Generalized Subgradients and Its Properties

Huang Zhenggang

College of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054

Received:2014-04-20 Revised:2015-03-04 Online:2015-10-25 Published:2015-10-25

摘要/Abstract

摘要：

该文提出了一种新的广义次梯度,讨论了与Tangent锥相关的一些基本性质,且在部分性质基础上得出了含不等式与任意集约束下非凸非可微标量最优化问题的k阶严格局部极小点存在的必要与充分条件以及无约束非凸非可微标量最优化问题局部极小点存在的必要条件.

关键词: k-下(上)次梯度, Tangent锥, k阶严格局部极小点, 局部极小点, 最优性条件

Abstract:

This paper proposes a new notion of generalized subgradients and discusses some properties of it in terms of Tangent cone. Using some of those properties, necessary and sufficient optimality conditions for a nonconvex and nondifferentiable scalar optimization problem with inequalities constraints and an arbitrary set constraint are derived, also a necessary condition for a nonconvex and nondifferentiable unconstrained scalar optimization problem is derived.

Key words: k-Lower(upper)subgradient, Tangent cone, Strict local minimizer of order k, Local minimizer, Optimality conditions

中图分类号:

O224

黄正刚. 一种新的广义次梯度及其性质[J]. 数学物理学报, 2015, 35(5): 927-935.

Huang Zhenggang. A New Notion of Generalized Subgradients and Its Properties[J]. Acta mathematica scientia,Series A, 2015, 35(5): 927-935.

参考文献

[1] Dini U. Fondamenti per la teoria delle funzioni di variabili reali. Pisa:Nistri, 1878
[2] Avriel M. Nonlinear Programming. Englewood Cliffs, NJ:Prentice-Hall, 1976
[3] Clarke F H. Generalized gradients and applications. Transactions of the American Mathematical Society, 1975, 205:247-262
[4] Clarke F H. Optimization and Nonsmooth Analysis(second edition). Philadelphia, PA:SIAM, 1990
[5] Michel P, Penot J-P. Calcul sous-differentiel pour les fonctions lipschiziennes et non lipschizienennes. C R Acad Sci Paris, 1984, 298:269-272
[6] Michel P, Penot J-P. A generalized derivative for calm and stable functions. Differential and Integaral Equations, 1992, 5(2):433-454
[7] Borwein J M, Zhu Q J. Techniques of Variational Analysis. New York:Springer, 2005
[8] Hestense M R. Calculus of Varitions and Optimal Control Theory. New York:Wiley, 1966
[9] Hestense M R. Optimization Theory. The Finite Dimensional Case. New York:Wiley, 1975
[10] Borwein J M. Convex Analysis and Nonlinear Optimization(Theory and Examples). New York:Springer, 2000
[11] Rockafellar R T. Convex Analysis. Princeton:Princeton University Press, 1970
[12] Bazaraa M B, Shetty C M. 非线性规划-理论与算法. 王化存, 张春柏译. 贵阳:贵州人民出版社, 1986
[13] Rockafellar R T, Wets R J-B. Variational Analysis. Berlin:Springer, 1998
[14] Modukhovich B S. Variational Analysi and Generalized Differentiation, I:Basic Theory, Grundlehren Series(Fundamental Principles of Mathematical Sciences). Berlin:Springer, 2006
[15] Schirotzek, W. Nonsmooth Analysis. Berlin:Springer, 2007
[16] Ferrera J. An Introduction to Nonsmooth Analysis. New York:Academic Press, 2013
[17] Kutateladze S S. Constructive Nonsmooth Analysis and Related Topics. New York:Springer, 2014
[18] Georgiev P G, Zlateva N P. Second-order subdifferentials of C1,1 functions and optimality conditions. Set-Valued Analysis, 1996, 4(2):101-117
[19] Studniarski M. Necessary and sufficient conditions for isolated local minima of nonsmooth functions. SIAM Journal on Control and Optimization, 1986, 24:1044-1049
[20] Jiménez B, Novo V. Higher-order optimality conditions for strict local minima. Annual of Operational and Research, 2008, 157:183-192