Generalized convexity in mathematical programming

Abstract

The role of convexity in the development of mathematical programming is reviewed. Many recent generalizations of convexity and their applications to optimization theory are discussed.

On generalised convex mathematical programming

“…When p = 1, the definition of V-invexity reduces to the notion of invexity in the sense of Hanson [9] with r](x, a) = a,(x, a)t\{x, a). When p = 1 and r\{x, a) = x -a, the condition reduces to strong pseudo-convexity condition (see [11,2]). The problem (VP) is said to be V-invex if the vector function (f y , ... , f p , g { , ... , g m ) is V-invex.…”

Strong pseudo-convexity and symmetric duality in nonlinear programming

“…It may be remarked here that strong Pseudo-convexity is not a modification of the usual pseudoconvexity, but rather is a special case of invex, as mentioned by Mond [7]. Thus…”

On generalized convexity and duality with a square root term

“…In the case of (~, p)-convexity assumptions, relative to (P), the work in [20] extends those due to Mond [15]. On the other hand, Chandra, Craven and Mond [3] study, in the spirit of Mond and Weir [191 duality theory for a mathematical programming problem with inequality constraints and objective function F. The advantage of duals presented in [16], [19] is that they allow the weakening of the usual convexity type hypothesis needed for Wolfe duality to hold. Thus, the Mond-Weir duality results from [3] are proved under generalized convexity assumptions, i.e., pseudoconvexity and quasiconvexity assumptions.…”