Higher-Order Minimizers and Generalized (F,ρ)-Convexity in Nonsmooth Vector Optimization over Cones

DOI: 10.4236/am.2015.61002   PDF   HTML   XML   3,053 Downloads   3,567 Views   Citations

Abstract

In this paper, we introduce the concept of a (weak) minimizer of order k for a nonsmooth vector optimization problem over cones. Generalized classes of higher-order cone-nonsmooth (F, ρ)-convex functions are introduced and sufficient optimality results are proved involving these classes. Also, a unified dual is associated with the considered primal problem, and weak and strong duality results are established.

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Suneja, S. , Sharma, S. and Kapoor, M. (2015) Higher-Order Minimizers and Generalized (F,ρ)-Convexity in Nonsmooth Vector Optimization over Cones. Applied Mathematics, 6, 7-19. doi: 10.4236/am.2015.61002.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Bector, C.R., Chandra, S. and Bector, M.K. (1988) Sufficient Optimality Conditions and Duality for a Quasiconvex Programming Problem. Journal of Optimization Theory and Applications, 59, 209-221.
[2] Mangasarian, O.L. (1969) Nonlinear Programming. McGraw-Hill, New York.
[3] Vial, J.P. (1983) Strong and Weak Convexity of Sets and Functions. Mathematics of Operations Research, 8, 231-259.
http://dx.doi.org/10.1287/moor.8.2.231
[4] Hanson, M.A. and Mond, B. (1982) Further Generalization of Convexity in Mathematical Programming. Journal of Information and Optimization Sciences, 3, 25-32.
http://dx.doi.org/10.1080/02522667.1982.10698716
[5] Preda, V. (1992) On Efficiency and Duality for Multiobjective Programs. Journal of Mathematical Analysis and Applications, 166, 365-377.
http://dx.doi.org/10.1016/0022-247X(92)90303-U
[6] Antczak, T. and Kisiel, K. (2006) Strict Minimizers of Order m in Nonsmooth Optimization Problems. Commentationes Mathematicae Universitatis Carolinae, 47, 213-232.
[7] Antczak, T. (2011) Characterization of Vector Strict Global Minimizers of Order 2 in Differentiable Vector Optimization Problems under a New Approximation Method. Journal of Computational and Applied Mathematics, 235, 4991-5000.
http://dx.doi.org/10.1016/j.cam.2011.04.029
[8] Cromme, L. (1978) Strong Uniqueness: A Far Criterion for the Convergence Analysis of Iterative Procedures. Numerische Mathematik, 29, 179-193.
http://dx.doi.org/10.1007/BF01390337
[9] Studniarski, M. (1989) Sufficient Conditions for the Stability of Local Minimum Points in Nonsmooth Optimization. Optimization, 20, 27-35.
http://dx.doi.org/10.1080/02331938908843409
[10] Bhatia, G. and Sahay, R.R. (2013) Strict Global Minimizers and Higher-Order Generalized Strong Invexity in Multiobjective Optimization. Journal of Inequalities and Applications, 2013, 31.
http://dx.doi.org/10.1186/1029-242X-2013-31
[11] Clarke, F.H. (1983) Optimization and Nonsmooth Analysis. Wiley, New York.
[12] Studniarski, M. (1997) Characterizations of Strict Local Minima for Some Nonlinear Programming Problems. Nonlinear Analysis, Theory, Methods & Applications, 30, 5363-5367.
http://dx.doi.org/10.1016/S0362-546X(97)00352-0
[13] Ward, D.W. (1994) Characterizations of Strict Local Minima and Necessary Conditions for Weak Sharp Minima. Journal of Optimization Theory and Applications, 80, 551-571.
http://dx.doi.org/10.1007/BF02207780
[14] Nahak, C. and Mohapatra, R.N. (2012) Nonsmooth -Invexity in Multiobjective Programming Problems. Optimization Letters, 6, 253-260.
http://dx.doi.org/10.1007/s11590-010-0239-1
[15] Craven, B.D. (1989) Nonsmooth Multiobjective Programming. Numerical Functional Analysis and Optimization, 10, 49-64.
http://dx.doi.org/10.1080/01630568908816290
[16] Cambini, R. and Carosi, L. (2010) Mixed Type Duality for Multiobjective Optimization Problems with Set Constraints. In: Jim?nez, M.A., Garzon, G.R. and Lizana, A.R., Eds., Optimality Conditions in Vector Optimization, Bentham Science Publishers, Sharjah, 119-142.

  
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