Some Results about Set-Valued Complementarity Problem

Document Type: research paper

Authors

Department of Mathematics, Faculty of Science, Golestan University, Gorgan, Iran

Abstract

This paper is devoted to consider the notions of complementary problem (CP) and set-valued complementary problem (SVCP). The set-valued complementary problem is compared with the classical single-valued complementary problem. Also, the solution set of the set-valued complementary problem is characterized. Our results illustrated by some examples.


This paper is devoted to consider the notions of complementary problem (CP) and set-valued complementary problem (SVCP). The set-valued complementary problem is compared with the classical single-valued complementary problem. Also, the solution set of the set-valued complementary problem is characterized. Our results illustrated by some examples. Our results illustrated by some examples.

This paper is devoted to consider the notions of complementary problem (CP) and set-valued complementary problem (SVCP). The set-valued complementary problem is compared with the classical single-valued complementary problem. Also, the solution set of the set-valued complementary problem is characterized. Our results illustrated by some examples. Our results illustrated by some examples.

Keywords


Article Title [Persian]

نتایجی پیرامون مسئله‌ی مکمل مجموعه-‌مقدار

Authors [Persian]

  • خدیجه بای‌پور
  • مهدی روحی
گروه ریاضی، دانشکده علوم، دانشگاه گلستان، گرگان، ایران
Abstract [Persian]

در این مقاله، ضمن مقایسه‌ی مسئله‌ی مکمل مجموعه-‌مقدار با مسئله‌ی مکمل تک‌مقدار، ساختار مجموعه‌ی جواب مسئله‌ی مکمل مجموعه-‌مقدار و شرایط لازم و کافی برای کرانداری و تحدب آن بررسی می‌شود. با معرفی رده‌ی جدیدی از ماتریس‌ها شرط لازم و کافی برای شدنی بودن مسئله‌ی مکمل خطی مجموعه-‌مقدار به‌دست می‌آید. همچنین چند کاربرد از مسئله ارائه می‌گردد.

Keywords [Persian]

  • مسئله‌ی مکمل
  • مسئله‌ی مکمل مجموعه-‌مقدار
  • مسئله‌ی مکمل تک‌-مقدار
  • مجموعه‌ی جواب

[1] J. P. Aubin and H. Frankowska, Set Valued Analysis, Birkhäuser, Boston, 1990.

 

[2] D. Aussel, N. Hadjisavvas, Onquasi-monotonevariationalinequalities, Journal of Optimization Theory and Applications. 121(2004), 445–450.

 

[3] H. H. Bauschke, and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, (2011).

 

[4] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63(1994), 123–145.

 

[5] R. W. Cottle, J. S. Pang and R. E. Stone, The Linear Complementarity Problem, Academic Press, NewYork, 1992.

 

[6] S. P. Dirkse, M. C. Ferris, MCPLIB : a collection of nonlinear rmixed complementarity problems, Optimization Methods & Software 5(1995), 319–345.

 

[7] B. C. Eaves, The linear complementa-rity problem, Management Science. 17(1971), 612–634.

 

[8] F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume I and II, Springer, New York, 2003.

 

[9] M. Fiedler and V. Ptak, Some generalizations of positive definiteness and monotonicity, Numerical Mathematics 9(1966), 163-172.

 

[10] M.S.Gowda, On the extended linear complementarity problem, Mathematical Programming 72(1996), 33–50.

[11] G. Isac, Topological Methods in Complementarity Theory, Nonconvex Optimization and Its Applications, Springer, 2000.

 

[12] O. L. Mangasarian and J. S. Pang, The extended linear complementarity problem, SIAM Journal on Matrix Analysis and Application 16(1995), 359–368.

 

[13] K .G. Murty, Linear Complementarity, Linear and Nonlinear Programming, Sigma Series in Applied Mathematics, Heldermann Verlag, Berlin, 1988.

 

[14] K. G. Murty, Supply chain management in the computerindustry, Manuscript, Department of IOE, Universityof Michigan, Ann Arbor, (1998).

 

[15] J. S. Pang, The implicit complementarity problem, in Nonlinear Programming, O. L. Mangasarian, R. R. Meyer, and S. M. Robinson, Eds., Academic Press, New York, (1981), 487–518.

 

[16] J. S. Pang and M. Fukushima, Quasi-variational inequalities, generalized Nash equilibria, and multileader-follower games, Computational Management Science. 2(2005), 21–56.

 

[17] E. Polak, Optimization: Algorithms and Consistent Approximation, Springer, New York, NY, USA, 1997.

 

[18] J. Zhou, J. S. Chen and G. M. Lee, On set-valued complementarity problems, Abstract and Applied Analysis, Vol. 2013, Article ID 105930, 11 pages, 2013.