In collaboration with Payame Noor University and the Iranian Society of Instrumentation and Control Engineers

Document Type : Research Article

Author

Department of Mathematics‎, ‎Payame Noor University, ‎P.O‎. ‎Box‎. ‎19395-3697‎, ‎Tehran‎, ‎Iran

Abstract

For a nonsmooth multiobjective mathematical programming problem governed by infinitely many constraints‎, ‎we define a new gap function that generalizes the definitions of this concept in other articles‎. ‎Then‎, ‎we characterize the efficient‎, ‎weakly efficient‎, ‎and properly efficient solutions of the problem utilizing this new gap function‎. ‎Our results are based on $(\Phi,\rho)-$invexity‎, ‎defined by Clarke subdifferential.

Keywords

[1] Antczak T. (2015) ”Saddle point criteria and Wolfe duality in nonsmooth (Φ, ρ)-invex vector optimization problems with inequality and equality constraints”, International J. Computer Math. 92, 882-907.
[2] Antczak T., Stasiak A. (2011) ”(Φ, ρ)-Invexity in Nonsmooth Optimization”, Numer. Func. Anal. Optim.32, 1-25.
[3] Ben-Israel A., Mond B. (1986) ”What is invexity?”, J. Australian Math. Sci. 28, 1-9.
[4] Caristi G., Kanzi M. and Soleimani-damaneh M. (2017) ”On gap functions for nonsmooth multiobjective optimization problems”, Optim Letters. DOI: 10.1007/s11590-017-1110-4.
[5] Caristi G., Ferrara M. and Stefanescu A. (2010) ”Semi-infinite multiobjective programming with generalized invexity”, Mathematical Reports, 62, 217-233.
[6] Caristi G., Ferrara M. and Stefanescu A. (2006) ”Mathematical programming with (ρ, Φ)- invexity”, In Generalized Convexity and Related Topics. Lecture Notes in Economics and Mathematical Systems, Vol. 583. (I.V. Konnor, D.T. Luc, and A.M. Rubinov, eds.). Springer, Berlin-Heidelberg-New York, 167-176.
[7] Caristi G., Kanzi N. (2015) ”Karush-Kuhn-Tuker Type Conditions for Optimality of NonSmooth Multiobjective Semi-Infinite Programming”, International Journal of Mathematical Analysis 9, 1929-1938.
[8] Clarke F. H. (1983) ”Optimization and Nonsmooth Analysis”, Wiley, Interscience.
[9] Chen C. Y., Goh C. J., Yang, X. Q. (1998) ”The gap function of a convex multicriteria optimization problem”, Eur. J. Oper. Res. 111, 142–151.
[10] Ehrgott M. (2005) ”Multicriteria Optimization”, Springer, Berlin.
[11] Gao X. Y. (2013) ”Optimality and duality for non-smooth multiobjective semi-infinite programming”, J. Netw. 8, 413-420.
[12] Goberna M.A., Kanzi N. (2017) ”Optimality conditions in convex multiobjective” SIP. Math. Programming. DOI 10.1007/s10107-016-1081-8.
[13] Goberna M. A., Guerra-Vazquez F. and Todorov M. I. (2016) ”Constraint qualifications in linear vector semi-infinite optimization”, European J. Oper. Res. 227, 32-40.
[14] Goberna M. A., Guerra-Vazquez F. and Todorov M. I. (2013) ”Constraint qualifications in convex vector semi-infinite optimization”, European J. Oper. Res. 249, 12-21.
[15] Gopfert A., Riahi H., Tammer C. and Zalinescu C. (2003) ”Variational methods in partial ordered spaces”, Springer, New York.
[16] Guerraggio A., Molho E. and Zaffaroni A. (1994) ”On the notion of proper efficiency in vector optimization”, J. Optim. Theory Appl. 82, 1-21.
[17] Hearn D. W., (1982) ”The gap function of a convex program”, Oper. Res. Lett. 1, 67–71.
[18] Kanzi N., Shaker Ardekani J., Caristi G. (2018) ”Optimality, scalarization and duality in linear vector semi-infinite programming”, Optimization. Doi: 10.1080/02331934.2018.1454921.
[19] Kanzi N. (2017) ”Necessary and sufficient conditions for (weakly) efficient of nondifferentiable multi-objective semi-infinite programming”, Iranian Journal of Science and Technology Transactions A: Science, DOI 10.1007/s40995-017-0156-6.
[20] Kanzi N. (2015) ”Karush-Kuhn-Tucker Types Optimality Conditions For Non-Smooth Semi-Infinite Vector Optimization Problems”, Journal of Mathematical Extension 9, 45- 56.
[21] Kanzi N. (2015) ”Regularity Conditions for Non-Differentiable Infinite Programming Problems Using Michel-Penot Subdifferential”,  Control and Optimization in Applied Mathematics, 1, 21-30.
[22] Noor M. A. (2004) ”On generalized preinvex functions and monotonicities”, Journal of Inequalities in Pure and Applied Mathematics. 5, 1–9.
[23] Rezayi A. (2018) ”Characterization of Isolated Efficient Solutions in Nonsmooth Multiobjective Semi-infinite Programming”, Iran J Sci Technol Trans Sci, Doi.org/10.1007/s40995- 018-0637-2.