Abstract
This note is concerned with the generalization of Farkas' theorem of the alternative and its application to derive the necessary optimality conditions for min-max problems with satisfaction conditions. Farkas' theorem is generalized to a system of linear inequalities with max operations. The problems studied require a solution at which the worst objective value attains its minimum over a set of solutions fulfilling satisfaction conditions. The satisfaction conditions claim that plural performance criteria should be kept below the permissible level, whatever disturbances may happen or whatever opponents' decisions may be taken. We present a generalized Farkas' theorem in order to derive the necessary optimality conditions for the problems of this class.
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References
Mangasarian, O. L.,Nonlinear Programming, McGraw-Hill Book Company, New York, New York, 1969.
Shimizu, K.,Systems Optimization Theory, Corona Publishing, Tokyo, Japan, 1976.
Gehner, K. R.,Necessary and Sufficient Optimality Conditions for the Fritz John Problem with Linear Equality Constraints, SIAM Journal on Control, Vol. 12, No. 1, 1974.
Schmitendorf, W. E.,Necessary Conditions and Sufficient Conditions for Static Min-Max Problems, Journal of Mathematical Analysis and Applications, Vol. 57, No. 3, 1977.
Shimizu, K., andAiyoshi, E.,A Theory and Algorithms for an Optimization Satisfaction Problem under Uncertainty, Institute of Electronics and Communication Engineers of Japan, Transactions, Vol. 62-A, No. 5, 1979.
Shimizu, K., andAiyoshi, E.,Necessary Conditions for Min-Max Problems and Algorithms by a Relaxation Procedure, IEEE Transactions on Automatic Control, Vol. AC-25, No. 1, 1980.
Danskin, J. M.,The Theory of Min-Max, with Applications, SIAM Journal on Applied Mathematics, Vol. 14, No. 4, 1966.
Blankenship, J. W., andFalk, J. E.,Infinitely Constrained Optimization Problems, Journal of Optimization Theory and Applications, Vol. 19, No. 2, 1976.
Bracken, J., andMcGill, J. T.,Defense Applications of Mathematical Programs with Optimization Problems in the Constraints, Operations Research, Vol. 22, No. 5, 1974.
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Communicated by G. Leitmann
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Shimizu, K., Aiyoshi, E. & Katayama, R. Generalized Farkas' theorem and optimization of infinitely constrained problems. J Optim Theory Appl 40, 451–462 (1983). https://doi.org/10.1007/BF00933510
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DOI: https://doi.org/10.1007/BF00933510