Generalized convexity in nondifferentiable programming

For an abstract mathematical programming problem involving quasidifferentiable cone-constraints we obtain necessary (and sufficient) optimality conditions of the Kuhn-Tucker type without recourse to a constraint qualification. This extends the known results to the non-differentiable setting. To obtain these results we derive several simple conditions connecting various concepts in generalized convexity not requiring differentiability of the functions involved.

[1]  R. Tyrrell Rockafellar.  Conjugate Duality and Optimization , 1974 .

[2]  B. M. Glover,et al.  Invex functions and duality , 1985, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[3]  C. R. Bector.  Programming Problems with Convex Fractional Functions , 1968, Oper. Res..

[4]  Olvi L. Mangasarian,et al.  Nonlinear Programming , 1969 .

[5]  M. Guignard.  Generalized Kuhn–Tucker Conditions for Mathematical Programming Problems in a Banach Space , 1969 .

[6]  F. Clarke.  Generalized gradients of Lipschitz functionals , 1981 .

[7]  S. Karamardian,et al.  Strictly quasi-convex (concave) functions and duality in mathematical programming , 1967 .

[8]  S. Yamamuro,et al.  Differential Calculus in Topological Linear Spaces , 1974 .

[9]  Sanjo Zlobec,et al.  Stabilization and determination of the set of minimal binding constraints in convex programming 1 , 1981 .

[10]  B. D. Craven,et al.  On minimization subject to cone constraints , 1983 .

[11]  R. Mifflin.  Semismooth and Semiconvex Functions in Constrained Optimization , 1977 .

[12]  M. Nashed.  Differentiability and Related Properties of Nonlinear Operators: Some Aspects of the Role of Differentials in Nonlinear Functional Analysis , 1971 .

[13]  Louis B. Rall,et al.  Nonlinear Functional Analysis and Applications , 1971 .

[14]  Adi Ben-Israel,et al.  Characterization of optimality in convex programming without a constraint qualification , 1976 .

[15]  Constantln Zǎlinescu.  On an abstract control problem , 1980 .

[16]  J. Hiriart-Urruty.  New concepts in nondifferentiable programming , 1979 .

[17]  Jonathan M. Borwein,et al.  Fractional programming without differentiability , 1976, Math. Program..

[18]  Kanti Swarup,et al.  Duality in fractional programming , 1968, Unternehmensforschung.

[19]  W. Ziemba,et al.  Generalized concavity in optimization and economics , 1981 .

[20]  Bevil Milton Glover,et al.  A generalized Farkas lemma with applications to quasidifferentiable programming , 1982, Z. Oper. Research.

[21]  G. A. Garreau,et al.  Mathematical Programming and Control Theory , 1979, Mathematical Gazette.

[22]  Jean-Baptiste Hiriart-Urruty,et al.  On optimality conditions in nondifferentiable programming , 1978, Math. Program..

[23]  B. D. Craven,et al.  Complete characterization of optimality for convex programming in banach spaces , 1980 .

[24]  G. Lebourg,et al.  Generic differentiability of Lipschitzian functions , 1979 .

[25]  Frank H. Clarke,et al.  A New Approach to Lagrange Multipliers , 1976, Math. Oper. Res..

[26]  B. N. Pshenichnyi.  Necessary Conditions for an Extremum , 1971 .

[27]  J. Borwein,et al.  Characterizations of optimality without constraint qualification for the abstract convex program , 1982 .