Abstract
In recent times the Kuhn—Tucker optimality conditions and the duality theorems for convex programming have been extended by generalizations of the convexity concept. In this paper the notion of a symmetric derivative for a function of several variables is introduced and used to provide extensions of some fundamental optimality and duality theorems of convex programming. Symmetric derivatives are also used to extend some optimality and duality theorems involving pseudoconvexity and differentiable quasiconvexity.
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References
K.J. Arrow and A.C. Enthoven, “Quasiconcave programming,”Econometrica 29 (1961) 779.
C.E. Aull, “The first symmetric derivative,”American Mathematical Monthly 74 (1967) 708.
J. Farkas, “Über die Theorie der einfachen Ungleichungen,”Journal für die reine und angewandte Mathematik 124 (1901) 1.
D.G. Luenberger, “Quasiconvex programming,”SIAM Journal of Applied Mathematics 16 (1968) 1000.
O.L. Mangasarian,Nonlinear programming (McGraw-Hill, New York, 1969).
R.A. Minch, “On generalizations of differentiability and convexity,” Ph.D. dissertation, Rensselaer Polytechnic Institute (1970).
T.S. Motzkin, “Beiträge zur Theorie der linearen Ungleichungen,” Dissertation, University of Basel, Jerusalem (1936).
M. Slater, “Lagrange multipliers revisited: a contribution to nonlinear programming,” Cowles Commission Discussion Paper, Mathematics 403 (Nov. 1950).
P. Wolfe, “A duality theorem for nonlinear programming,”Quarterly of Applied Mathematics 19 (1961) 239.
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Minch, R.A. Applications of symmetric derivatives in mathematical programming. Mathematical Programming 1, 307–320 (1971). https://doi.org/10.1007/BF01584095
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DOI: https://doi.org/10.1007/BF01584095