On augmented Lagrangians for Optimization Problems with a Single Constraint

On asymptotic Lagrangian duality for nonsmooth optimization

“…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]. Most of these papers share the following three features: (i) generate a primal-dual sequence at each iteration, (ii) under mild assumptions, prove that every accumulation point of the primal sequence is a solution of (P) , and (iii) the dual problem is convex.…”

Existence and characterization theorems in nonconvex vector optimization

“…One of the main application areas is the nonconvex vector optimization, where these functions were used to characterize efficient solutions (see e.g., [7,14,17,22,30,33]). Another application area of these functions is the single objective mathematical programming (see e.g., [11][12][13][25][26][27]) where the conical supporting surfaces were used to develop optimality conditions and algorithms for calculating optimal solutions. The paper is organized as follows.…”

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.…”