Abstract
Some results of approximation type for multiobjective optimization problems with a finite number of objective functions are presented. Namely, for a sequence of multiobjective optimization problems P n which converges in a suitable sense to a limit problem P, properties of the sequence of approximate Pareto efficient sets of the P n 's, are studied with respect to the Pareto efficient set of P. The exterior penalty method as well as the variational approximation method appear to be particular cases of this framework.
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References
Aubin, J. P. (1979), Mathematical Methods of Game and Economic Theory, North Holland Publ. Amsterdam.
Attouch, H. (1984), Variational Convergence for Functions and Operators, Pitman Advanced Publishing Program, Boston-London-Melbourne.
Aze, D. and J. P., Penot (1987), Operations on Convergent Families of Sets and Functions, Publication AVAMAC, Université de Perpignan, France, no. 87–05, Vol. 1.
Céa, J. (1971), Optimisation, théorie et algorithmes, Dunod.
Dugośiia, D. P. and M. D., Aśić (1986), Uniform Convergence and Pareto Optimality, Optimization 17 (6), 723–729.
Karlin, S. (1960), Mathematical Methods and Theory in Games, Programming and Economics, Vol. I, McGraw Hill, New York.
Kutateladze, S. S. (1979), Convex ε-Programming, Soviet Mathematical Doklady 20, 391–393.
Lemaire, B. (1988), Coupling Optimization Methods and Variational Convergence, International Series of Numerical Mathematics, Vol. 84(c), Birkhäuser Verlag, pp. 163–179.
Lemaire, B. (1987), Convergence in Vector Optimization. Communication at the Congrès Franco-Québécois d' Analyse Non-Linéaire, Perpignan, France.
Vály, István (1987), Epsilon Solutions and Duality in Vector Optimization, Working paper WP-87–43. Institute for Applied Systems Analysis. Laxenbourg, Austria.
White, D. J. (1984), Multiobjective Programming and Penalty Functions, JOTA 43 (4), 583–599.
White, D. J. (1986), Epsilon Efficiency, Journal of Optimization Theory and Application 49 (2), 319–337.
Zangwill, W. I. (1969), Nonlinear Programming, Prentice-Hall, Englewood Cliffs, New Jersey.
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Lemaire, B. Approximation in multiobjective optimization. J Glob Optim 2, 117–132 (1992). https://doi.org/10.1007/BF00122049
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DOI: https://doi.org/10.1007/BF00122049
Key words
- Epiconvergence
- ε-efficient set
- Mosco-convergence
- multiobjective optimization
- Pareto
- penalization
- variational approximation