A generalized Farkas lemma with applications to quasidifferentiable programming

Abstract

For sublinear mappings between normed linear spaces a generalization of Farkas' lemma is established thus extending the known results to include a class of nonqinear functions. A generalized Motzkin alternative theorem is established and as an application we provide a Kuhn-Tucker type necessary optimality condition for a non-differentiable mathematical programming problem.

Invex functions and duality

“…Since int 5 ^ 0 , there is a weak* compact convex set B c Y' such that [7 ] Invex functions and duality 7…”

“…As a consequence of Theorem 2 (to follow) we will establish the converse to part (b) above (see Remark 3), thus if every stationary point of/ 0 is a minimum then/ 0 is invex on X o . If the cone J x is not assumed weak* closed, then the Kuhn-Tucker conditions (KT) may be replaced by the doubly asymptotic Kuhn-Tucker conditions (see [25], [18], [7])…”

“…It is possible to consider Fritz John type conditions in an asymptotic form (see [7]) which would be applicable when J x is not necessarily closed. The conditions (FJ) are known to be satisfied when the cone S has non-empty (topological) interior ( [3]).…”

“…Theorem 4 generalizes the result of Hanson [10] and Craven [5] given in Theorem l(a). The condition (12) has been shown to be necessary for optimality by Glover [7,Theorem 4] under the quasidifferentiability assumptions of Theorem 4 and the additional hypotheses that / 0 is arc-wise directionally differentiable at a ( [6]) and g is locally solvable at a. In the special case of Theorem 4 in which f 0 and g are linearly Gateaux differentiable at a it is easily shown that (12) is equivalent to (AKT).…”

“…We can consider generalized Fritz John conditions (under suitable regularity and quasidifferentiability assumptions (see [7]) for problems (P) to attain a minimum at a G E; namely Thus, analogously to Remark 3, part (ii), / is ^0-invex at a on E if and only if either {GFJ + ) is no/ satisfied at a G E, or, the corresponding Lagrangean function attains a minimum at a over £. This result follows easily from (22); we need only assume J' x is weak* closed for each x G E.…”