The Zero Duality Gap Property and Lower Semicontinuity of the Perturbation Function

Abstract

We examine the validity of the zero duality gap properties for two important dual schemes: a generalized augmented Lagrangian dual scheme and a nonlinear Lagrange-type dual scheme. The necessary and sufficient conditions for the zero duality gap property to hold are established in terms of the lower semicontinuity of the perturbation functions.

On asymptotic Lagrangian duality for nonsmooth optimization

“…because (19) shows that the infimum value q k j (λ 2 ) cannot be attained over the set X(r 0 ) . The above expression contradicts (14), and hence asymptotic duality holds.…”

“…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].…”

Abstract Convexity and Augmented Lagrangians

“…Augmented Lagrangians with a nonconvex augmenting function have been intensively studied as well (see [9,17,8,19,6,23,24,25] and references therein). Some of these references (e.g., [9,17,6,19]) use abstract convexity tools in their analysis.…”

“…Some of these references (e.g., [9,17,6,19]) use abstract convexity tools in their analysis. Our aim is: (i) to present a unified analysis for the examination of nonconvex augmented Lagrangians for a wider family of augmenting terms, and (ii) express the main definitions and facts describing the augmented Lagrangian theory in terms of abstract convexity tools.…”

“…One of these families consists of the "linear" or "ordinary" augmenting terms, and the other provides the "penalty" terms. Our resulting augmenting term includes, as a particular case, the ones studied in [13,17,8,23,25].…”