A general Farkas lemma and characterization of optimality for a nonsmooth program involving convex processes

In this paper, a generalization of the Farkas lemma is presented for nonlinear mappings which involve a convex process and a generalized convex function. Using this result, a complete characterization of optimality is obtained for the following nonsmooth programming problem: minimizef(x), subject to − ∈H(x) wheref is a locally Lipschitz function satisfying a generalized convexity hypothesis andH is a closed convex process.

[1]  C. Swartz.  The Farkas lemma of Shimizu, Aiyoshi and Katayama , 1985, Bulletin of the Australian Mathematical Society.

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

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

[4]  Vaithilingam Jeyakumar,et al.  Nonlinear alternative theorems and nondifferentiable programming , 1984, Z. Oper. Research.

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

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

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

[8]  A. Kruger.  CONDITIONS FOR AN EXTREMUM , 1986 .

[9]  C. Swartz,et al.  A general Farkas lemma , 1985 .

[10]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[11]  R. Holmes.  Geometric Functional Analysis and Its Applications , 1975 .

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

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

[14]  R. Rockafellar.  Conjugate Duality and Optimization , 1987 .

[15]  Constantin Zălinescu,et al.  A generalization of the Farkas lemma and applications to convex programming , 1978 .

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

[17]  Jonathan M. Borwein,et al.  Adjoint Process Duality , 1983, Math. Oper. Res..