Nonlinear separation approach for the augmented Lagrangian in nonlinear semidefinite programming

Augmented Lagrangian functions for cone constrained optimization: the existence of global saddle points and exact penalty property

“…where [·] + denotes the projection of a matrix onto the cone S l + (see [78,76,96,77,86,57,87,90,92] for more details on this augmented Lagrangian). One can check that the function Φ(y, λ, c) defined above is a particular case of the function Φ(y, λ, c) from Example 1 with σ(y) = ( y 0 2…”

“…The axiomatic approach that we use was inspired by [50]. It provides one with a simple and unified framework for the study of various augmented Lagrangian functions, such as the Hestenes-Powell-Rockafellar augmented Lagrangian [2,66,67], the cubic augmented Lagrangian [42], Mangasarian's augmented Lagrangian [59,89], the exponential penalty function [1,80,79,50,83], the Log-Sigmoid Lagrangian [64,65], the penalized exponential-type augmented Lagrangians [1,79,50,83], the modified barrier functions [63,79,50,83], the p-th power augmented Lagrangian [43,44,91,46,45,88,50], He-Wu-Meng's augmented Lagrangian [36], extensions of the Hestenes-Powell-Rockafellar augmented Lagrangian to the case of nonlinear second order cone programs [51,52,97], nonlinear semidefinite programs [40,78,76,96,77,86,57,87,90,92], and semi-infinite programs [70,…”

“…The first part of the paper is devoted to the problem of the existence of global saddle points of augmented Lagrangian functions, which is important for convergence analysis of augmented Lagrangian methods, since the existence of a global saddle point is, usually, necessary for the global convergence of these methods. The problem of the existence of global saddle points was studied for general cone constrained optimization problems in [72,99], for mathematical programming problems in [50,49,56,98,89,82,79,81], for nonlinear second order cone programming problems in [97], for nonlinear semidefinite programming problems in [90,55], and for semi-infinite programming problems in [70,6]. The analysis of the known results on the existence of global saddle points of augmented Lagrangian functions indicates that the same, in essence, results are proved and reproved multiple times in different settings.…”

Existence of augmented Lagrange multipliers: reduction to exact penalty functions and localization principle

“…where [·] + denotes the projection of a matrix onto the cone of m × m positive semidefinite matrices. In order to obtain necessary and sufficient conditions for the existence of an augmented Lagrange multiplier for the problem (22), let us recall KKT optimality conditions for this problem [35,41]. Let x * be a locally optimal solution of the problem (22), and the functions f 0 , G and h be twice differentiable at x * .…”

Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points

“…Particularly, Pareto/weak efficient solution notions of vector optimization have been extensively studied; see [1,2,6,3,32,27]. In recent years, optimality conditions for optimization problems are widely studied since they play a crucial role in duality theory and algorithm design; see [1,7,8,9,10,28,33,34]. Basically, all optimality conditions are derived using theorem of separation and theorem of alterative or an adequate substitute such as Ekeland's principle; see [6,7,34,11,12,13].…”