GLOBAL OPTIMIZATION OF THE DIFFERENCE OF TWO INCREASING PLUS-CONVEX-ALONG-RAYS FUNCTIONS

doi:10.1007/s10473-020-0615-6

数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (6): 1849-1873.doi: 10.1007/s10473-020-0615-6

GLOBAL OPTIMIZATION OF THE DIFFERENCE OF TWO INCREASING PLUS-CONVEX-ALONG-RAYS FUNCTIONS

H. SHAHRIARIPOUR¹, H. MOHEBI²

1. Department of Mathematics, Graduate University of Advanced Technology, Kerman, Iran;
2. Department of Mathematics and Mahani Mathematical Research Center, Shahid Bahonar University of Kerman, P. O. Box:76169133, Postal Code:7616914111, Kerman, Iran

收稿日期:2019-06-06 修回日期:2020-07-17 出版日期:2020-12-25 发布日期:2020-12-30
通讯作者: H. MOHEBI,E-mail:hmohebi@uk.ac.ir E-mail:hmohebi@uk.ac.ir
作者简介:H. SHAHRIARIPOUR,E-mail:hakimeh-shahriari@yahoo.com
基金资助:
The second author was partially supported by the Mahani Mathematical Research Center, Iran, grant no:97/3267.

GLOBAL OPTIMIZATION OF THE DIFFERENCE OF TWO INCREASING PLUS-CONVEX-ALONG-RAYS FUNCTIONS

H. SHAHRIARIPOUR¹, H. MOHEBI²

1. Department of Mathematics, Graduate University of Advanced Technology, Kerman, Iran;
2. Department of Mathematics and Mahani Mathematical Research Center, Shahid Bahonar University of Kerman, P. O. Box:76169133, Postal Code:7616914111, Kerman, Iran

Received:2019-06-06 Revised:2020-07-17 Online:2020-12-25 Published:2020-12-30
Contact: H. MOHEBI,E-mail:hmohebi@uk.ac.ir E-mail:hmohebi@uk.ac.ir
Supported by:
The second author was partially supported by the Mahani Mathematical Research Center, Iran, grant no:97/3267.

摘要/Abstract

摘要： The theory of increasing and convex-along-rays (ICAR) functions defined on a convex cone in a real locally convex topological vector space X was already well developed. In this paper, we first examine abstract convexity of increasing plus-convex-along-rays (IPCAR) functions defined on a real normed linear space X. We also study, for this class of functions, some concepts of abstract convexity, such as support sets and subdifferentials. Finally, as an application, we characterize the maximal elements of the support set of strictly IPCAR functions and give optimality conditions for the global minimum of the difference between two IPCAR functions.

关键词: increasing plus-convex-along-rays function, support set, maximal element, global minimum, DC-function

Abstract: The theory of increasing and convex-along-rays (ICAR) functions defined on a convex cone in a real locally convex topological vector space X was already well developed. In this paper, we first examine abstract convexity of increasing plus-convex-along-rays (IPCAR) functions defined on a real normed linear space X. We also study, for this class of functions, some concepts of abstract convexity, such as support sets and subdifferentials. Finally, as an application, we characterize the maximal elements of the support set of strictly IPCAR functions and give optimality conditions for the global minimum of the difference between two IPCAR functions.

Key words: increasing plus-convex-along-rays function, support set, maximal element, global minimum, DC-function

中图分类号:

90C46

H. SHAHRIARIPOUR, H. MOHEBI. GLOBAL OPTIMIZATION OF THE DIFFERENCE OF TWO INCREASING PLUS-CONVEX-ALONG-RAYS FUNCTIONS[J]. 数学物理学报(英文版), 2020, 40(6): 1849-1873.

H. SHAHRIARIPOUR, H. MOHEBI. GLOBAL OPTIMIZATION OF THE DIFFERENCE OF TWO INCREASING PLUS-CONVEX-ALONG-RAYS FUNCTIONS[J]. Acta mathematica scientia,Series B, 2020, 40(6): 1849-1873.

参考文献

[1] Billera L J. On games without side payments arising from a general class of markets. J Math Econom, 1974, 1(2):129-139
[2] Daryaei M H, Mohebi H. Abstract convexity of extended real valued ICR functions. Optim, 2013, 62:835-855
[3] Doagooei A R, Mohebi H. Optimization of the difference of topical functions. J Global Optim, 2013, 57:1349-1358
[4] Doagooei A R, Mohebi H. Optimization of the difference of ICR functions. Nonlinear Anal, 2009, 71:449-499
[5] Doagooei A R, Mohebi H. Monotonic analysis over ordered topological vector spaces:IV. J Global Optim, 2009, 45:355-369
[6] Dutta J, Martínez-Legaz J -E, Rubinov A M. Monotonic analysis over cones:I Optim, 2004, 53:165-177
[7] Dutta J, Martínez-Legaz J -E, Rubinov A M. Monotonic analysis over cones:II. Optim, 2004, 53:529-547
[8] Eberhard A C, Mohebi H. Maximal abstract monotonicity and generalized Fenchel's conjugation formulas. Set-Valued Anal, 2010, 18:79-108
[9] Gaubert S, Gunawardena J. A non-linear hierarchy for discrete event dynamical systems//Proceedings of the 4th Workshop on Discrete Event Systems, Calgiari, Technical Report HPL-BRIMS-98-20, HewlettPackard Labs. Cambridge:Cambridge University Press, 1998
[10] Gunawardena J. An introduction to idempotency. Idempotency (Bristol, 1994), 1-49, Publ Newton Inst, 11. Cambridge:Cambrige Univ Press, 1998
[11] Gunawardena J. From max-plus algebra to non-expansive mappings:a nonlinear theory for discrete event systems. Theoretical Computer Science, to appear (revised version of technical Report HPL-BRIMS-99-07, Hewlett-Packard Labs, 1999)
[12] Gunawardena J, Keane M. On the existence of cycle times for some non-expansive maps. Technical Report HPL-BRIMS-95-003, Hewlett-Packard Labs, 1995
[13] Jeyakumar J, Rubinov A M, Zhiyou W. Generalized Fenchel's conjugation formulas and duality for abstract convex functions. J Optim Theory Appl, 2007, 132:441-458
[14] Levin V L. Abstract convexity in measure theory and in convex analysis. J Math Sci, 2003, 4:3432-3467
[15] Levin V L. Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich problem. Set-Valued Anal, 1999, 1:7-32
[16] Martínez-Legaz J -E. Quasiconvex duality theory by generalized conjugation methods. Optim, 1988, 19:1029-4945
[17] Mohebi H, Samet M. Abstract convexity of topical functions. J Global Optim, 2014, 58:365-357
[18] Mohebi H, Esmaeili M. Optimization of the difference of increasing and radiant functions. Appl Math Sci, 2012, 67:3339-3345
[19] Mohebi A, Mohebi H. Global optimization of the difference of affine increasing and positively homogeneous functions. Optim, 2016, 65(12):2081-2093
[20] Mohebi H, Sadeghi H. Monotonic analysis over non-convex cones. Numer Funct Anal Optim, 2005, 26:879-895
[21] Mohebi H, Sadeghi H. Monotonic analysis over ordered topological vector spaces:I. Optim, 2007, 56:305-321
[22] Nedíc A, Ozdaglar A, Rubinov A M. Abstract convexity for nonconvex optimization duality. Optim, 2007, 56:655-674
[23] Penot J -P. Autoconjugate functions and representations of monotone operators. Bull Austral Math Soc, 2003, 67:277-284
[24] Rockafellar R T. Convex Analysis. Princeton, NJ:Princeton University Press, 1970
[25] Renner P, Schmedders K. A polynomial optimization approach to principal-agent problems. Econometrica, 2015, 83(2):729-769
[26] Rubinov A M, Gasimov R N. Strictly increasing positively homogeneous functions with application to exact penalization. Optim, 2003, 52(1):1-28
[27] Rubinov A M. Abstract convexity, global optimization and data classification. OPSEARCH, 2001, 38(3):247-265
[28] Rubinov A M, Andramono M Y. Minimizing increasing star-shaped functions based on abstract convexity. J Global Optim, 1999, 15(1):19-39
[29] Rubinov A M, Glover B M. Increasing convex-along-rays functions with applications to global optimization. J Optim Theory Appl, 1999, 102(3):615-642
[30] Rubinov A M, Vladimirov A. Convex-along-rays functions and star-shaped sets. Numer Funct Anal Optim, 1998, 19(5/6):593-613
[31] Rubinov A.M., Uderzo A. On global optimality conditions via separation functions. J Optim Theory Appl, 2001, 109:345-370
[32] Rubinov A M, Wu Z Y. Optimality conditions in global optimization and their applications. Math Program, Ser B, 2009, 120:101-123
[33] Rubinov A M, Dzalilov Z. Abstract convexity of positively homogeneous functions. J Statist Manag Syst, 2002, 5:1-20
[34] Rubinov A M. Abstract Convexity and Global Optimization. Boston Dordrecht London:Kluwer Academic Publishers, 2000
[35] Rubinov A M. Some properties of increasing convex-along-rays functions. Proc Centre Math, Appl Austral Nat Univ, 1999, 36:153-167
[36] Rubinov A M, Singer I. Topical and sub-topical functions, downward sets and abstract convexity. Optim, 2001, 50:307-351
[37] Sharkey W W. Convex games without side payments. Internat J Game Theory, 1981, 10(2):101-106
[38] Shveidel A P. Abstract convex sets with respect to the class of general min-type functions. Optim, 2003, 52:571-579
[39] Singer I. On radiant sets, downward sets, topical functions and sub-topical functions in lattice ordered groups. Optim, 2004, 53:393-428
[40] Singer I. Abstract Convex Analysis. New York:Wiley-Interscience, 1997
[41] Yao C, Li S. Vector topical functions, abstract convexity and image space analysis. J Optim Theory Appl, 2018, 177:717-742
[42] Zalinescu C. Convex Analysis in General Vector Spaces. London:World Scientific, 2002

GLOBAL OPTIMIZATION OF THE DIFFERENCE OF TWO INCREASING PLUS-CONVEX-ALONG-RAYS FUNCTIONS

GLOBAL OPTIMIZATION OF THE DIFFERENCE OF TWO INCREASING PLUS-CONVEX-ALONG-RAYS FUNCTIONS

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 4

Metrics

本文评价

推荐阅读 0

[1]	丁协平. NEW SYSTEMS OF GENERALIZED QUASI-VARIATIONAL INCLUSIONS IN FC-SPACES AND APPLICATIONS[J]. 数学物理学报(英文版), 2011, 31(3): 1142-1154.
[2]	丁协平. MATHEMATICAL PROGRAMS WITH SYSTEM OF GENERALIZED VECTOR QUASI-EQUILIBRIUM CONSTRAINTS IN FC-SPACES[J]. 数学物理学报(英文版), 2010, 30(4): 1257-1268.
[3]	丁协平. SYSTEM OF GENERALIZED VECTOR QUASI-EQUILIBRIUM PROBLEMS ON PRODUCT FC-SPACES[J]. 数学物理学报(英文版), 2007, 27(3): 522-534.
[4]	丁协平. COINCIDENCE THEOREMS INVOLVING COMPOSITES OF ACYCLIC MAPPINGS AND APPLICATIONS[J]. 数学物理学报(英文版), 1999, 19(1): 53-61.