Duality for a Non‐Convex Program in a Real Banach Space

Abstract

AbstractThis paper works out a direct duality theorem for a mathematical program in Banach space having ratio of a convex and a linear functional as its objective function. For a convex program in real BANACH space, RITTER [1] extended the WOLFE's duality theorem for a convex program in EUCLIDEAN space. It however turns out that WOLFE's direct duality theorem does not hold for a pseudoconvex program in general [2, p. 158]. BECTOR [3] succeeded in proving WOLFE's direct duality theorem for a mathematical program in EUCLIDEAN space in which the pseudo‐convex objective function is the ratio of a convex and a linear function. The present paper aims at extending this result for programs in BANACH space. The approach to establish the direct duality theorem, however, is the same as that of RITTER, i.e. using the weak duality theorem and the Kuhn‐Tucker necessary conditions for optimality.