Abstract
In this study, the optimization theory of Dubovitskii and Milyutin is extended to multiobjective optimization problems, producing new necessary conditions for local Pareto optima. Cones of directions of decrease, cones of feasible directions and a cone of tangent directions, as well as, a new cone of directions of nonincrease play an important role here. The dual cones to the cones of direction of decrease and to the cones of directions of nonincrease are characterized for convex functionals without differentiability, with the aid of their subdifferential, making the optimality theorems applicable. The theory is applied to vector mathematical programming, giving a generalized Fritz John theorem, and other applications are mentioned. It turns out that, under suitable convexity and regularity assumptions, the necessary conditions for local Pareto optima are also necessary and sufficient for global Pareto optimum. With the aid of the theory presented here, a result is obtained for the, so-called, “scalarization” problem of multiobjective optimization.
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References
M. S. Bazaraa andJ. J. Goode, Necessary optimality criteria in mathematical programming in the presence of differentiability,J. Math. Analysis & App. 40 (1972), 609–621.
A. Ben-Israel, Linear equations and inequalities on finite dimensional, real or complex, vector spaces: a unified theory,J. Math. Analysis & App. 27 (1969), 367–389.
Y. Censor,Contributions to Optimization Theory: Multiobjective Problems, doctoral thesis, Technion, Haifa, 1975 (Hebrew).
N. O. Da Cunha andE. Polak, Constrained minimization under-vector-valued criteria in finite dimensional spaces,J. Math. Analysis & App. 19 (1967), 103–124.
N. O. Da Cunha and E. Polak,Constrained Minimization Under-Vector-Valued Criteria in Linear Topological Spaces, in Math. Theory of Control, Academic Press, Eds: Balakrishnan, A. V., and Newstadt, L. W., 96–108, 1967.
V. F. Demyanov and A. M. Rubinov,Approximate Methods in Optimization Problems, American-Elsevier Pub. Co., 1970.
A. Y. Dubovitskii andA. A. Milyutin, Extremu problems with constraints,Dokl. Akad. Nauk. SSSR,149 (1963), 759–762. (English Translation,Soviet. Math. Dolk. 4 (1963), 452–455.)
A. Y. Dubovitskii andA. A. Milyutin, Extremum problems in the presence of constraints,Zh. Vychist. Mat. i Mat. Fiz. 5 (1965), 395–453. (English translation:USSR Comp. Math. and Math. Physics 5 (1965) 1–80.)
A. M. Geoffrion, Proper efficiency and the theory of vector maximization,J. Math. Analysis & Appl. 22 (1968), 618–630.
I. V. Girsanov,Lectures on Mathematical Theory of Extremum Problems, Lecture Notes in Economics and Mathematical Systems, #67, Springer-Verlag, Berlin, Heidelberg, New York, 1972.
P-J. Laurent,Approximation et Optimisation, Hermann, Paris, 1972.
J. Ponstein, Seven Kinds of Convexity,SIAM Review 9 (1967), 115–119.
B. N. Pshenichnyi,Necessary Conditions for an Extremum, Marcel Dekker Inc., 1971.
R. T. Rockafellar,Convex Analysis, Princeton Univ. Press, 1970.
S. Smale, Global Analysis and Economics V,J. Math Economics 1 (1974), 213–221.
J. Stoer andC. Witzgall,Convexity and Optimization in Finite Dimensions I, Springer-Verlag, Berlin, Heidelberg, New York, 1970.
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Communicated by J. Stoer
The author's work in this area is now supported by NIH grants HL 18968 and HL 4664 and NCI contract NO1-CB-5386.
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Censor, Y. Pareto optimality in multiobjective problems. Appl Math Optim 4, 41–59 (1977). https://doi.org/10.1007/BF01442131
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DOI: https://doi.org/10.1007/BF01442131