Abstract Convexity and Augmented Lagrangians

Abstract

The ultimate goal of this paper is to demonstrate that abstract convexity provides a natural language and a suitable framework for the examination of zero duality gap properties and exact multipliers of augmented Lagrangians. We study augmented Lagrangians in a very general setting and formulate the main definitions and facts describing the augmented Lagrangian theory in terms of abstract convexity tools. We illustrate our duality scheme with an application to stochastic semi-infinite optimization.

Semi-continuous quadratic optimization: existence conditions and duality scheme

“…Proof Given the perturbation function φ of f , define the marginal function h as in (15). Thus, the Fenchel-Moreau conjugate of the marginal function h is given by h * :…”

“…In this section, we apply the general duality scheme, that can be found in [10,15,18], in the setting of SCM problems with a particular choice for the coupling function. Without any loss of generality, we can redefine f considering, in addition, f (x) = +∞ when x / ∈ X , and, equivalently, rewrite the SCM problem (5) as…”

“…In fact, the Fenchel-Moreau conjugation has been already studied in a general manner in [10,[14][15][16][17][18], and it is implicit in [11]. However, here, we apply the general scheme specifically for the SCM problems.…”

On asymptotic Lagrangian duality for nonsmooth optimization

“…The following additional assumption on σ is often used in the context of augmented Lagrangians [11,20].…”

“…Indeed, it has been shown that for a certain family of augmented Lagrangians, both zero duality gap and saddle point properties hold [18,Chapter 11]. These duality properties have been extended to more general kinds of Lagrangians and more general frameworks (including infinite dimensional spaces) [11,13,17,19,20]. Many solution techniques for nonconvex optimization rely on the good duality properties of augmented Lagrangians [2,3,4,5,6,7,8,9,14].…”