The infinite dimensional Lagrange multiplier rule for convex optimization problems

Abstract

In this paper an infinite dimensional generalized Lagrange multipliers rule for convex optimization problems is presented and necessary and sufficient optimality conditions are given in order to guarantee the strong duality. Furthermore, an application is presented, in particular the existence of Lagrange multipliers associated to the bi-obstacle problem is obtained.

On the Study of an Economic Equilibrium with Variational Inequality Arguments

“…Moreover, from Propositions 2.2 and 2.3, ∂(−u a ) is a maximal monotone and strongly monotone map. Hence, by Theorem 2.3, there exists a unique solution to GVI (12). So, we can define the function:…”

“…So, there exists at least one subsequence {x a,n k } such that x a,n k → t ∈ R l . (ii) t is the solution to (12)…”

“…If we replace x a,n = x a,n in (15) and we pass to the limit, then one has u a (x a ) ≤ u a (t). That is t is the unique solution to GVI (12): t = x a ( p).…”

“…Later on, in order to study some problems from operations research, mathematical programming and optimization theory, Chan and Pang in [9] generalized the quasi-variational inequality considered in [8], by introducing the so-called generalized quasi-variational inequality. In [10][11][12][13][14][15] different equilibrium problems are studied by means of variational inequalities.…”

Existence and Approximation for Variational Problems Under Uniform Constraints on the Gradient by Power Penalty

“…Ekeland and Temam (1976) show the insufficiency of the traditional duality theory for tackling this problem. The question was solved positively by Daniele et al (2007) using a new infinite dimensional duality theory (see also Donato, 2011;Maugeri and Puglisi, 2014). Daniele et al (2007) show, for a large class of problems including Problems (2.1) and (1.2), that if the problem is solvable and the solution satisfies a constraint qualification condition, then there exists a Lagrange multiplier λ ∈ L ∞ + satisfying (1.3b), which is indeed the solution of a dual problem.…”