Continuous time non-smooth optimization through quasi efficiency

Kumar, Promila; Sharma, Bharti

doi:10.2478/candc-2023-0038

Abstract

The importance of quasi efficiency lies in its versatile nature as it permits a definite tolerable error that depend on the decision variables. This has been a motivating factor for us to introduce the notion of quasi e cient solution for the non-smooth multiobjective continuous time programming problem. Necessary optimality conditions are derived for this problem. To derive sufficient optimality conditions, the concept of approximate convexity has been extended to continuous case in this paper. A mixed dual is proposed for which weak and strong duality results are proved.

References

Bellman, R. (1953) Bottleneck problems and dynamic programming. Proc. Natl. Acad. Sci. USA 39, 947–951.
Search in Google Scholar Back to article
Bhatia, D., Gupta A. and Arora, P. (2013) Optimality via generalized approximate convexity and quasiefficiency. Optimization Letters 7, 127–135.
Search in Google Scholar Back to article
Brandao, A. J. V., Rojas-Medar M. A. and Silva, G. N. (1998) Non- smooth continuous-time optimization problems: sufficient conditions. J. Math. Anal. Appl. 227, 305–318.
Search in Google Scholar Back to article
Brandao, A. J. V., Rojas-Medar, M. A. and Silva, G. N. (2001) Nonsmooth continuous-time optimization problems: necessary conditions. Comput. Math. Appl. 41, 1477–1486.
Search in Google Scholar Back to article
Chuong, T. D. and Kim, D. S. (2016) Approximate solutions of multiobjective optimization problems. Positivity 20, 187–207.
Search in Google Scholar Back to article
Clarke, F. H. (1983) Optimization and Non-Smooth Analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley and Sons, Inc., New York.
Search in Google Scholar Back to article
Farr, W. H. and Hanson, M. A. (1974) Continuous-time programming with nonlinear constraints. J. Math. Anal. Appl. 45, 96–115.
Search in Google Scholar Back to article
Golestani, M., Sadeghi, H. and Tavan, Y. (2018) Nonsmooth multiobjective problems and generalized vector variational inequalities using quasi efficiency. J. Optim. Theory Appl. 179, 896–916.
Search in Google Scholar Back to article
Gupta, A., Mehra A. and Bhatia, D. (2006) Approximate convexity in vector optimization. Bull. Austral. Math. Soc. 74, 207–218.
Search in Google Scholar Back to article
Loridan, P. (1982) Necessary Conditions for ǫ-Optimality. Math Program. 19, 140–152.
Search in Google Scholar Back to article
Loridan, P. (1984) ǫ-solutions in vector minimization problems. J. Optim. Theory Appl. 43, 265–276.
Search in Google Scholar Back to article
Mishra, S. K. and Upadhyay, B. B. (2013) Some relations between vector variational inequality problems and nonsmooth vector optimization problems using quasi efficiency. Positivity 17, 1071–1083.
Search in Google Scholar Back to article
Nobakhtian S. and Pouryayevali, M. R. (2008a) Optimality criteria for nonsmooth multiobjective continuous-time problems. J. Optim. Theory Appl. 136, 69–76.
Search in Google Scholar Back to article
Nobakhtian, S. and Pouryayevali, M. R. (2008b) Duality for nonsmooth continuous-time problems of vector optimization. J. Optim. Theory Appl. 136, 77–85.
Search in Google Scholar Back to article
Reiland, T. W. (1980) Optimality conditions and duality in continuous programming I. Convex programs and a theorem of alternative. J. Math. Anal. Appl. 77, 329–343.
Search in Google Scholar Back to article
Reiland, T. W. and Hanson, M. A. (1980) Generalized Kuhn-Tucker conditions and duality for continuous nonlinear programming problems. J. Math. Anal. Appl. 74, 578–598.
Search in Google Scholar Back to article
Upadhyay, B. B., Stancu-Minasian, I. M. and Mishra, P. (2023) On relations between nonsmooth interval-valued multiobjective programming problems and generalized Stampacchia vector variational inequalities. Optimization 72, 2635-2659.
Search in Google Scholar Back to article
Upadhyay, B. B., Mishra, P., Mohapatra, R. N. and Mishra, S. K. (2019) On the applications of nonsmooth vector optimization problems to solve generalized vector variational inequalities using convexificators. Adv. Intell. Syst. Comput. 991. DOI: 10.1007/978-3-030-21803-466.
Search in Google Scholar Back to article
Zalmai, G. J. (1985a) Optimality conditions and Lagrangian duality in continuous-time nonlinear programming. J. Math. Anal. Appl. 109, 426–452.
Search in Google Scholar Back to article
Zalmai, G. J. (1985b) The Fritz John and Kuhn-Tucker optimality conditions in continuous-time programming. J. Math. Anal. Appl. 110, 503–518.
Search in Google Scholar Back to article

Continuous time non-smooth optimization through quasi efficiency

Abstract

Paradigm

My account