In this work, by using weak conjugate maps given in (Azimov and Gasimov, in Int J Appl Math 1:171–192, <CitationRef CitationID="CR1">1999</CitationRef>), weak Fenchel conjugate dual problem, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${(D_F^w)}$$</EquationSource> </InlineEquation> , and weak Fenchel Lagrange conjugate dual problem <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${(D_{FL}^w)}$$</EquationSource> </InlineEquation> are constructed. Necessary and sufficient conditions for...</equationsource></inlineequation></equationsource></inlineequation></citationref>

In this paper we show that for every nonempty convex compact subset K of a finite dimensional space E there exists a Lipschitz function F: E → R, such that ∂F, generalized gradient of F in the sense of Clarke [4] is equal everywhere to K.

This paper constructs a closed set Y in Rl such that for all y in the boundary of Y, Clarke's normal cone to Y at y is equal to Rl+. If Y is the production set of a firm, then the marginal cost pricing rule imposes no restriction. The existence of Y is shown to be equivalent to the existence of...