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Optimality and duality with generalized convexity. (English) Zbl 0838.90114
Summary: Hanson and Mond have given sets of necessary and sufficient conditions for optimality and duality in constrained optimization by introducing classes of generalized convex functions, called type I and type II functions. Recently, Bector defined univex functions, a new class of functions that unifies several concepts of generalized convexity. In this paper, optimality and duality results for several mathematical programs are obtained combining the concepts of type I and univex functions. Examples of functions satisfying these conditions are given.
MSC:
90C30Nonlinear programming
26B25Convexity and generalizations (several real variables)
90C32Fractional programming
90C29Multi-objective programming; goal programming
References:
[1]Hanson, M. A.,On Sufficiency of the Kuhn-Tucker Conditions, Journal of Mathematical Analysis and Applications, Vol. 80, pp. 545–550, 1981. · Zbl 0463.90080 · doi:10.1016/0022-247X(81)90123-2
[2]Hanson, M. A., andMond, B.,Necessary and Sufficient Conditions in Constrained Optimization, Mathematical Programming, Vol. 37, pp. 51–58, 1987. · Zbl 0622.49005 · doi:10.1007/BF02591683
[3]Bector, C. R., andSingh, C.,B-Vex Functions, Journal of Optimization Theory and Applications, Vol. 71, pp. 237–253, 1991. · Zbl 0793.90069 · doi:10.1007/BF00939919
[4]Bector, C. R., Suneja, S. K., andLalitha, C. S.,Generalized B-Vex Functions and Generalized B-Vex Programming, Proceedings of the Administrative Sciences Association of Canada, pp. 42–53, 1991.
[5]Bector, C. R., Suneja, S. K., andGupta, S.,Univex Functions and Univex Nonlinear Programming, Proceedings of the Administrative Sciences Association of Canada, pp. 115–124, 1992.
[6]Hanson, M. A., andMond, B.,Further Generalizations of Convexity in Mathematical Programming, Journal of Information and Optimization Sciences, Vol. 3, pp. 25–32, 1982.
[7]Singh, C.,Duality Theory in Multiobjective Differentiable Programming, Journal of Information and Optimization Sciences, Vol. 9, pp. 231–240, 1988.
[8]Singh, C., andHanson, M. A.,Multiobjective Fractional Programming Duality Theory, Naval Research Logistics, Vol. 38, pp. 925–933, 1991.
[9]Bector, M. K., Husain I., Chandra, S., andBector, C. R.,A Duality Model for a Generalized Minmax Program, Naval Research Logistics, Vol. 35, pp. 493–501, 1988. · doi:10.1002/1520-6750(198810)35:5<493::AID-NAV3220350512>3.0.CO;2-R