Abstract
Necessary Kuhn-Tucker conditions up to precision ɛ without constraint qualification for ɛ-Pareto optimality of multiobjective programming are derived. This article suggests the establishment of a Wolfe-type ɛ-duality theorem for nondifferentiable, nonconvex, multiobjective minimization problems. The ɛ-vector Lagrangian and the generalized ɛ-saddle point for Pareto optimality are studied.
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References
Kutateladze, S. S.,Convex ɛ-Programming, Soviet Mathematics Doklady, Vol. 20, pp. 391–393, 1979.
Loridan, P.,Necessary Conditions for ɛ-Optimality, Mathematical Programming Study, Vol. 19, pp. 140–152, 1982.
Loridan, P.,ɛ-Solution in Vector Minimization Problems, Journal of Optimization Theory and Applications, Vol. 43, pp. 265–267, 1984.
Loridan, P., andMorgan, J.,Penalty Functions in ɛ-Programming and ɛ-Minimax Problems, Mathematical Programming, Vol. 26, pp. 213–231, 1983.
Strodiot, J. J., Nguyen, V. H., andHeukemes, M.,ɛ-Optimal Solutions in Nondifferentiable Convex Programming and Some Related Questions, Mathematical Programming, Vol. 25, pp. 307–308, 1983.
Clarke, F. H.,A New Approach to Lagrange Multipliers, Mathematics of Operations Research, Vol. 1, pp. 165–174, 1976.
Clarke, F. H.,Generalized Gradient and Applications, Transactions of the American Mathematical Society, Vol. 205, pp. 247–262, 1975.
Ekeland, I.,On the Variational Principle, Journal of Mathematical Analysis and Applications, Vol. 47, pp. 324–353, 1974.
Additional bibliography
Censor, Y.,Pareto Optimality in Multiobjective Problems, Applied Mathematics and Optimization, Vol. 4, pp. 41–59, 1979.
Kanniappan, P.,Necessary Conditions for Optimality of Nondifferentiable Convex Multiobjective Programming, Journal of Optimization Theory and Applications, Vol. 40, pp. 167–174, 1983.
Kanniappan, P., andSastry, S. M. A.,Duality Theorems and an Optimality Condition for Nondifferentiable Convex Programming, Journal of the Australian Mathematical Society, Series A, Vol. 32, pp. 369–379, 1982.
Lai, H. C., andLiu, J. C.,A Necesary and Sufficient Condition of Convex Multiobjective Programming, Tamkang Journal of Mathematics, Vol. 20, pp. 7–17, 1989.
Lai, H. C., andHo, C. P.,Duality Theorem of Nondifferentiable Convex Multiobjective Programming, Journal of Optimization Theory and Applications, Vol. 50, pp. 407–420, 1986.
Minami, M.,Weak Pareto Optimality of Multiobjective Problems in a Locally Convex Linear Topological Space, Journal of Optimization Theory and Applications, Vol. 34, pp. 469–484, 1981.
Minami, M.,Weak Pareto Optimality of Multiobjective Problem in a Banach Space, Bulletin of Mathematical Statistics, Vol. 19, pp. 19–23, 1981.
Minami, M.,Weak Pareto-Optimal Necessary Conditions in a Nondifferentiable Multiobjective Program on a Banach Space, Journal of Optimization Theory and Applications, Vol. 41, pp. 451–461, 1983.
Mond, B., andZlobec, S.,Duality for Nondifferentiable Programming without a Constraint Qualification, Utilitas Mathematica, Vol. 15, pp. 291–302, 1979.
Nieuwenhuis, J. W.,Some Minimax Theorems in Vector-Valued Functions, Journal of Optimization Theory and Applications, Vol. 40, pp. 463–475, 1983.
Schechter, M.,A Subgradient Duality Theorem, Journal of Mathematical Analysis and Applications, Vol. 61, pp. 850–855, 1977.
Schechter, M.,More on Subgradient Duality, Journal of Mathematical Analysis and Applications, Vol. 71, pp. 251–262, 1979.
Tanino, T., andSawaragi, Y.,Duality Theory in Multiobjective Programming, Journal of Optimization Theory and Applications, Vol. 27, pp. 509–529, 1979.
Wolfe, P.,A Duality Theorem for Nonlinear Programming, Quarterly of Applied Mathematics, Vol. 19, pp. 239–244, 1961.
Jahn, J.,Duality in Vector Optimization, Mathematical Programming, Vol. 25, pp. 343–353, 1983.
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Liu, J.C. ɛ-duality theorem of nondifferentiable nonconvex multiobjective programming. J Optim Theory Appl 69, 153–167 (1991). https://doi.org/10.1007/BF00940466
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DOI: https://doi.org/10.1007/BF00940466