Sufficient optimality conditions and duality for nonsmooth multiobjective optimization problems via higher order strong convexity

Authors

  • Balendu B. Upadhyay Department of Mathematics National Institute of Technology, Manipur, India
  • Ningthoujam Priyobarta Department of Mathematics National Institute of Technology, Manipur, India
  • Yumnam S. Rohen Department of Mathematics National Institute of Technology, Manipur, India

DOI:

https://doi.org/10.2298/YJOR170119013U

Keywords:

nonsmooth multiobjective programming, support functions, strict minimizers, optimality conditions, mixed duality

Abstract

In this paper, we define some new generalizations of strongly convex functions of order m for locally Lipschitz functions using Clarke subdifferential. Suitable examples illustrating the non emptiness of the newly defined classes of functions and their relationships with classical notions of pseudoconvexity and quasiconvexity are provided. These generalizations are then employed to establish sufficient optimality conditions for a nonsmooth multiobjective optimization problem involving support functions of compact convex sets. Furthermore, we formulate a mixed type dual model for the primal problem and establish weak and strong duality theorems using the notion of strict efficiency of order m. The results presented in this paper extend and unify several known results from the literature to a more general class of functions as well as optimization problems.

References

Arora, P., Bhatia, G., and Gupta, A., "Characterizations of the solution sets and sufficient optimality criteria via higher-order strong convexity", In: S. K. Mishra (ed.) Topics in Nonconvex Optimization: Theory and Applications, pp. 231-242. Springer Science + Business Media, LLC, New York, 2011.

Auslender, A., "Stability in mathematical programming with nondifferentiable data", SIAM Journal on Control and Optimization, 22 (1984) 239-254.

Bae, K.D., Kang, Y.M., and Kim, D.S., "Efficiency and generalized convex duality for nondifferentiable multiobjective programs", Journal of Inequalities and Applications, doi:10.1155/2010/930457 (2010).

Bae, K.D., and Kim, D.S., "Optimality and duality theorems in nonsmooth multiobjective optimization", Fixed Point Theory and Applications, doi:10.1186/1687-1812-2011-42 (2011).

Bhatia, G., "Optimality and mixed saddle point criteria in multiobjective optimization", Journal of Mathematical Analysis and Applications, 342 (2008) 135-145.

Clarke, F.H., "Optimization and Nonsmooth Analysis", John Wiley and Sons, New York, 1983.

Chandra, S., Dutta, J., and Lalitha, C.S., "Regularity conditions and optimality in vector optimization", Numerical Functional Analysis and Optimization, 25 (2004) 479-501.

Cromme, L., "Strong uniqueness: A for reaching criterion for the convergence of iterative numerical procedures", Numerical Mathematics, 29 (1978) 179-193.

Jimenez, B., "Strict efficiency in vector optimization", Journal of Mathematical Analysis and Applications, 265 (2002) 264-284.

Jimenez, B., and Novo, V., "First and second order sufficient conditions for strict minimality in multiobjective programming", Numerical Functional Analysis and Optimization, 23 (2002) 303-322.

Jimenez, B., and Novo, V., "First and second order sufficient conditions for strict minimality in nonsmooth vector optimization", Journal of Mathematical Analysis and Applications, 284 (2003) 496-510.

Karamardian, S., and Schaible, S., "Seven Kinds of Monotone Maps", Journal of Optimization Theory and Applications, 66 (1990) 37-46.

Kim, D.S., and Bae, K.D., "Optimality condition and duality for nondifferentiable multiobjective programs", Taiwanese Journal of Mathematics, 13 (2009) 789-804.

Kim, D.S., and Lee, H.J., "Optimality conditions and duality in nonsmooth multiobjective programs", Journal of Inequalities and Applications, doi:10.1155/2010/939537 (2010).

Lin, G.H., and Fukushima, M., "Some exact penalty results for non-linear programs and mathematical programs with equilibrium constraints", Journal of Optimization Theory and Applications, 118 (2003) 67-80.

Loe, A.D., "Approximate subdifferential and applications I: The finite dimensional theory", Transactions of American Mathematical Society, 281 (1984) 389-416.

Michel, P., and Penot, J.P., "Calculsous-differentiel pour de fonctions Lipschitziennes et nonlipschitziennes", C. R. Academy of Sciences Paris Series I Mathematics, 12 (1984) 269–272.

Mishra, S.K., and Upadhyay, B.B., "Some relation between vector variational inequality problems and nonsmooth vector optimization problems using quasi efficiency", Positivity, 17 (2013) 1071-1083.

Mishra, S.K., and Upadhyay, B.B., "Duality in nonsmooth multiobjective fractional programming involving pseudolinear functions", Indian Journal of Industrial and Applied Mathematics, 3 (2012) 152–161.

Mishra, S.K., and Upadhyay, B.B., "Efficiency and duality in nonsmooth multiobjective fractional programming involving pseudolinear functions", Yugoslav Journal of Operations Research, 22 (2012) 3–18.

Mishra, S.K., and Upadhyay, B.B., "Nonsmooth minimax fractional programming involving pseudolinear functions", Optimization, 63 (2014) 775–788.

Mishra, S.K., Upadhyay, B.B., and Hoai An, L.T., "Lagrange multiplier characterizations of solution sets of constrained nonsmooth pseudolinear optimization problems", Journal of Optimization Theory and Applications, 160 (2014) 763–777.

Mishra, S.K., and Upadhyay, B.B., "Pseudolinear Functions and Optimization", CRC Press, 2014.

Mishra, S.K., Wang, S.Y., and Lai, K.K., V-Invex Functions and Vector Optimization, Springer Science + Business Media, LLC, New York, 2008.

Mishra, S.K., Wang, S.Y., and Lai, K.K., Generalized Convexity and Vector Optimization, Springer Verlag, Berlin, Heidelberg, 2009.

Mond, B., and Schechter, M., "Nondifferentiable symmetric duality", Bulletin of the Australian Mathematical Society, 53 (1996) 177–187.

Mordukhovich, B.S., Variational Analysis and Generalized Differentiation I: Basic Theory, Springer, Berlin, 2006.

Rockafellar, R.T., Convex Analysis, Princeton University Press, Princeton, NJ, 1970.

Studniarski, M., "Necessary and sufficient conditions for isolated local minima of nonsmooth functions", SIAM Journal on Control and Optimization, 24 (1986) 1044–1049.

Ward, D.E., "Characterization of strict local minima and necessary conditions for weak sharp minima", Journal of Optimization Theory and Applications, 80 (1994) 551–557.

Yang, X.M., Teo, K.L., and Yang, X.Q., "Duality for a class of nondifferentiable multiobjective programming problems", Journal of Mathematical Analysis and Applications, 252 (2000) 999—1005.

Zalinescu, C., "On the differentiability of the support function", Journal of Global Optimization, 57 (2013) 719–731.

Downloads

Published

2017-05-01

Issue

Vol. 27 No. 2 (2017)

Section

Research Articles

License

Copyright (c) 2017 YUJOR

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.