Optimality Conditions and Constraint Qualifications for Quasiconvex Programming
- 29 Downloads
In mathematical programming, various kinds of optimality conditions have been introduced. In the research of optimality conditions, some types of subdifferentials play an important role. Recently, by using Greenberg–Pierskalla subdifferential and Martínez-Legaz subdifferential, necessary and sufficient optimality conditions for quasiconvex programming have been introduced. On the other hand, constraint qualifications are essential elements for duality theory in mathematical programming. Over the last decade, necessary and sufficient constraint qualifications for duality theorems have been investigated extensively. Recently, by using the notion of generator, necessary and sufficient constraint qualifications for Lagrange-type duality theorems have been investigated. However, constraint qualifications for optimality conditions in terms of Greenberg–Pierskalla subdifferential and Martínez-Legaz subdifferential have not been investigated yet. In this paper, we study optimality conditions and constraint qualifications for quasiconvex programming. We introduce necessary and sufficient optimality conditions in terms of Greenberg–Pierskalla subdifferential, Martínez-Legaz subdifferential and generators. We investigate necessary and/or sufficient constraint qualifications for these optimality conditions. Additionally, we show some equivalence relations between duality results for convex and quasiconvex programming.
KeywordsQuasiconvex programming Optimality condition Constraint qualification Generator of a quasiconvex function
Mathematics Subject Classification90C26 90C46 49J52
The author is grateful to anonymous referees for many comments and suggestions which improved the quality of the paper.
- 1.Boţ, R.I.: Conjugate Duality in Convex Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 637. Springer, Berlin, (2010)Google Scholar
- 13.Avriel, M., Diewert, W.E., Schaible, S., Zang, I.: Generalized concavity. Math. Concepts Methods Sci. Engrg. Plenum Press, New York (1988)Google Scholar
- 14.Ivanov, V.I.: Characterizations of solution sets of differentiable quasiconvex programming problems. J. Optim. Theory Appl. https://doi.org/10.1007/s10957-018-1379-1
- 25.Jeyakumar, V., Dinh, N., Lee, G. M.: A new closed cone constraint qualification for convex optimization. Research Report AMR 04/8, Department of Applied Mathematics, University of New South Wales, (2004)Google Scholar
- 40.Martínez-Legaz, J.E.: A generalized concept of conjugation. Lecture Notes in Pure and Appl. Math. 86, 45–59 (1983)Google Scholar
- 41.Martínez-Legaz, J.E.: A new approach to symmetric quasiconvex conjugacy. Lecture Notes in Econom. and Math. Systems. 226, 42–48 (1984)Google Scholar