The first order condition for composite optimization problems based onreformulationof the augmented Lagrangian

Optimization problems with abstract constraints given in a form of set inclusions have attracted a great deal of attention in recent years. This kind of optimization problems provides a unified framework for several important optimization problems, such as nonlinear programming, semidefinite programming, and second-order cone programming. Augmented Lagrangian play a key role in primal-dual methods for solving nonlinear programming. In this paper, we study the first order condition for the existence of Lagrange multipliers are presented for composite optimization problems. Analysis is based on Augmented Lagrangian formulation of this function. We also study the relations between if first order condition and this new Augmented Lagrangian multiplier

Keywords:

Augmented Lagrange multiplier , Moreau envelope , Firs-order condition