Duality in Semi-Infinite Programs and Some Works of Haar and Carathéodory

Abstract

By constructing a new infinite dimensional space for which the extreme point—linear independence and opposite sign theorems of Charnes and Cooper continue to hold, and, building on a little-known work of Haar (herein presented), an extended dual theorem comparable in precision and exhaustiveness to the finite space theorem is developed. Building further on this a dual theorem is developed for arbitrary convex programs with convex constraints which subsumes in principle all characterizations of optimality or duality in convex programming. No differentiability or constraint qualifications are involved, and the theorem lends itself to new computational procedures.

A dual characterization of incentive efficiency

“…nðPÞ and nðDÞ are finite). 7 However, unlike an ordinary linear program, a bounded LSIP problem need not have optimal solutions. Moreover, the primal and dual problems need not have the same optimal value, as a ''positive duality gap'' may occur: nðPÞ À nðDÞ40: The next two theorems show that the problems in this paper are well-behaved.…”

“…Following Charnes et al [7], we define the restricted dual problem, so-called dual problem in Haar's sense. The LSIP problems in Sections 2 and 3 obtain as particular cases of the problems in this section by applying the definitions in Table 1.…”

Geometric fundamentals of the simplex method in semi-infinite programming

“…Some of the conditions given in Theorem 3.1 are not new in the literature. Thus, (11) appeared in [2], whereas (III) and (V1) were introduced in [23] in a more general setting.…”

Farkas-Minkowski systems in semi-infinite programming

“…We suppose now that a'x>fl is a consequence relation of (a~x>=flt, tET}. For this purpose it is enough to prove that the solutions of the latter system of the form [-1]' [1] and [0]' are solutions of the relation (I). …”