Two characterizations of Pareto minima in convex multicriteria optimization. (English) Zbl 0549.90085

Summary: We give two conditions, each of which is both necessary and sufficient for a point to be a global Pareto minimum. The first one is obtained by studying programs where each criterion appears as a single objective function, while the second one is given in terms of a ”restricted Lagrangian”. The conditions are compared with the familiar characterization of properly efficient and weakly efficient points of Karlin and Geoffrion.


90C31 Sensitivity, stability, parametric optimization
90C25 Convex programming
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[1] R. Abrams L. Kerzner: A simplified test for optimality. Journal of Optimization Theory and Applications 25 (1978), 161-170. · Zbl 0352.90047
[2] A. Ben-Israel: Linear equations and inequalities on finite dimensional, real or complex, vector spaces: A unified theory. Journal of Mathematical Analysis and Applications 27 (1969), 367-389. · Zbl 0174.31502
[3] A. Ben-Israel A. Ben-Tal A. Charnes: Necessary and sufficient conditions for Pareto optimum in convex programming. Econometrica 45 (1977), 811 - 820. · Zbl 0367.90093
[4] A. Ben-Israel A. Ben-Tal S. Zlobec: Optimality in Nonlinear Programming: A Feasible Directions Approach. Wiley-Interscience, New York, 1981. · Zbl 0454.90043
[5] A. Ben-Tal A. Ben-Israel S. Zlobec: Characterization of optimality in convex programming without a constraint qualification. Journal cf Optimization Theory and Applications 20 (1976), 417-437. · Zbl 0327.90025
[6] G. R. Bitran T. L. Magnanti: The structure of admissible points with respect to cone dominance. Journal of Optimization Theory and Applications 29 (1979), 473 - 514. · Zbl 0389.52021
[7] Y. Censor: Pareto optimality in multiobjective problems. Applied Mathematics and Optimization 4 (1977), 41 - 59. · Zbl 0346.90055
[8] A. Charnes W. W. Cooper: Management Models and Industrial Applications of Linear Programming. Vol. I. Wiley, New York, 1961. · Zbl 0107.37004
[9] A. M. Geoffrion: Proper efficiency and the theory of vector maximization. Journal of Mathematical Analysis and Applications 22 (1968), 618 - 630. · Zbl 0181.22806
[10] S. Karlin: Mathematical Methods and Theory in Games. Programming and Economics. Vol. I, Addison-Wesley, Reading, Massachussetts, 1959. · Zbl 0139.12704
[11] V. V. Podinovskii: Applying the procedure for maximizing the basic local criterion to solving the vector optimization problems. Systems Control 6, Novosibirsk, 1970
[12] V. V. Podinovskii V. M. Gavrilov: Optimization with Respect to Successive Criteria. Soviet Radio, Moscow, 1975
[13] R. T. Rockafellar: Convex Analysis. Princeton University Press, 1970. · Zbl 0193.18401
[14] M. E. Salukvadze: Vector-Valued Optimization Problems in Control Theory. Academic Press, New York, 1979. · Zbl 0471.49001
[15] S. Smale: Global analysis and economics III. Journal of Mathematical Economics 1 (1974), 107-117. · Zbl 0316.90007
[16] S. Smale: Global analysis and economics V. Journal of Mathematical Economics 1 (1974), 213-221. · Zbl 0357.90010
[17] S. Smale: Global analysis and economics VI. Journal of Mathematical Economics 3 (1976), 1-14. · Zbl 0348.90017
[18] S. Zlobec: Regions of stability for ill-posed convex programs. Aplikace Matematiky 27 (1982), 176-191. · Zbl 0482.90073
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