Enlargements of Abstract Monotone Operators Determind by Representing Functions
محورهای موضوعی : مجله بین المللی ریاضیات صنعتی
H. Eshaghi Kenari
1
(Department of Mathematics and Statics, University of Birjand, Birjand, Iran.)
A. Asadi
2
(Department of Mathematics and Statics, University of Birjand, Birjand, Iran.)
MR. Miri
3
(Department of Mathematics and Statics, University of Birjand, Birjand, Iran.)
کلید واژه: Abstract monotone operator, Enlargement, ε- Abstract sub-differential, Abstract sub-differential,
چکیده مقاله :
In this paper, we study a new enlargement of abstract sub-differential for any IPH function. This nice property, apart from its theoretical importance, gives also the possibility to use the enlargement of abstract sub-differential in finding approximate solutions of inclusions determined by abstract sub-differentials. We define a new enlargement and observe, in the case abstract sub-differential, the relation between this new enlargement and the ε -abstract sub-differential.
در این مقاله ، ما یک افزایش جدید زیر دیفرانسیل انتزاعی برای هر عملکرد IPH را مطالعه می کنیم. ما می دانیم که ϵ-انتزاعی زیر دیفرانسیل از هر تابع IPH بزرگ شدن خلاصه ای از آن زیر دیفرانسیل و هر نقطه از نمودار ε-انتزاعی است. زیر دیفرانسیل را می توان با استفاده از نقط تقسیم زیر دیفرانسیل به نقطه تقریبی تبدیل کرد. این ویژگی خوب ، به غیر از اهمیت نظری آن ، امکان استفاده از بزرگنمایی دیفرانسیل زیر انتزاعی را نیز در محلولهای تقریبی اجزاء تعیین شده توسط تفاوتهای فرعی انتزاعی فراهم می آورد. ما یک پیشرفت جدید را مشاهده نمی کنیم و در مورد زیر دیفرانسیل انتزاعی ، رابطه بین این بزرگنمایی جدید و زیر دیفرانسیل ϵ-انتزاعی را مشاهده می کنیم.
[1] A. Brendsted, R. T. Rockafellar, On the sub-di erential of convex functions, Proc. Amer. Math. Soc. 16 (1965) 605-611.
[2] R. S. Burachick, A. M.Rubinov, On abstract convexity and set valued analysis, Journal of Nonlinear and Convex Analysis 9 (2009) 105-123.
[3] R. S. Barachik, C. A. Sagastizabal, B. F. Svaiter, Enlarqement of maximal monotone operatory: theory and Application, in Non-smooth piecewise Smooth, Semismooth and smoothing methods, M.fukushima and L.Qi (eds), Kluwer Academic publishers, (1998).
[4] R. S. Burachik, A. N. Iusem, B. F. Svaiter, Enlargements of maximal monotone operators with application to varitional inequali-ties, Set- valued Anal 5 (1997) 159-180.
[5] R. S. Burachick, B. F. Svaiter, Maximal monotone operators, convex functions and a special family of enlargements, Set Valued Analysis 10 (2002) 297-316.
[6] R. S. Burachick, B. F. Svaiter, Maximal monotonicity, conjugation and the duality product, Proc. Amer. Math. Soc. 131 (2003) 2379-238.
[7] S. Fitzpatrick, Representing monotone op- erators by convex functions, in Workahop and Miniconference on functional Analysis and Oprimization, Canberra, Austral. Nat. Univ., Canberra 12 (1988) 59-65.
[8] V. Jeyakumar, Q. M. Rubinov, Z. Y. Wu, Generalized Fenchels conjugation for mulas and duality for abstract convex functions, J. Optlm. Theory Appl. 132 (2007) 441-458.
[9] E. Kraus, Arepresentation of maximal monotone operations by saddle functions, Rev. Roum. Math. Pures Appl 30 (1985) 823-836.
[10] V. L. Levin, Abstract convexity in measute theory and in convex analysis, J. Math. Sci. 4 (2003) 3432-3467.
[11] V. L. Levin, Abestract cyclical monotonoicity and Monye solutions for the general Monge Kantorouich problem, set valued Analysis, Set valued Analysis 1 (1999) 7-32.
[12] R. Lucchetti, F. Patrone, A characterization of tykhonov well-posedness for minimum problem with application to variational inequalitiones number, Function. Anal. Optin. 4 (1987) 461-476.
[13] J. E. Martine-Legaz, B. Svaiter, Monotone operators representable by l.s.c.functions, Set Valued Analysis 13 (2005) 21-46.
[14] Y. E. Martines-Lagas, M. Thera, ϵ-subdi erentials in terms of subdi erential, Set-valued Anal 4 (1996) 327-332.
[15] J. E. Martine-Legaz, M. Thera, A convex representation of maximal monotone operators, Journal of Nonlinear and Convex Analysis 2 (2001) 243-247.
[16] J. J. Moreau, Inf-convolution, sousadditivite, convexite des fonctions numeriques, J. Math. Pures Appl. 49 (1970) 109-154.
[17] D. Pallaschke, S. Rolewicz, Foundations of mathematical optimization, Kluwer Academic Publishers, Boston, Dordrecht, London, (1997).
[18] J-P. Penot, The relevance of convex analysis for the study of monotonicity, Nonlinear Analysis 58 (2004) 855-871.
[19] R. T. Rockafellar, Local boundeness of nonlinear monotone operators, Michigan Math. 18 (1969) 397-407.
[20] R. T. Rockafellar, On the maximal monotonicity of subdi erential mappings, Paci c Journal of Mathematics 33 (1970) 209-216.
[21] A. M. Rubinov, Abstract convexity and global optimization, Kluwer Academic publishers, Boston, Dordrecht, London, (2000).
[22] A. M. Rubinov, M. Y. Andramonov, Minirnizing increasing star-shaped functions based on abstract convexity, Journal of Global Optimization 15 (1999) 19-39.
[23] A. M. Rubinov, B. M. Glover, Increasing convex-along-rays functions with applications to global optimization, Journal of optimization theory and Applications 102 (1999) 615-542
[24] A. M. Rubonov, A. Uderzo, On global optimality conditions via separations functions, Journal of Optimization Theory and Applications 109 (2001) 345-370.
[25] A. M. Rubinov, A. Vladimirov, Convexalong-rays functions and star-shaped sets, Numerical functional Analysis and Optimization 19 (1998) 593-613.
[26] B. F. Svaiter, Afamily of enlargements of maximal monotone operators, Set-Valued Anal. 8 (2000) 311-328.
[27] L. singer, Abstract convex analysis, WileyInterscience, New York, (1997).
[28] S. Simons, Minimax and monotonicity, Lectures Notes in Mathematics, Springer, New York, (1998).
[29] S. Dimons, C. Zalinescu, Fenchel duality, Fitzpatrick functions and maximal monotonicity, Journal of Nonlinear and convex Analysis 6 (2005) 1-22.
