Complexity and performance of an Augmented Lagrangian algorithm

Abstract

Algencan is a well established safeguarded Augmented Lagrangian algorithm introduced in [R.

Complexity of Proximal augmented Lagrangian for nonconvex optimization with nonlinear equality constraints

“…By using the bound a ≤ b + c =⇒ a 2 ≤ 2b 2 + 2c 2 for positive scalars a, b, c, and using the definition (10), we obtain the result.…”

“…where D S max{ x − y | x, y ∈ S 0 α} and C 1 , C 2 are defined as in (10). Then the following statements are true.…”

“…There are few results in the literature on outer iteration complexity in the nonconvex setting. Some quite recent results appear in [13,10]. In [13], the authors apply a general version of augmented Lagrangian to nonconvex optimization with both equality and inequality constraints.…”

“…Total iteration complexity measures are obtained for the case of linear equality constraints when the subproblem is solved with a p-order method (p ≥ 2). In [10], the authors studied an augmented Lagrangian framework named Algencan to problems with equality and inequality constraints. An ǫ-accurate first order point (whose precise definition is again similar to our ǫ-1o optimality in the case of equality constraints only) is obtained in O(| log ǫ|) outer iterations when the penalty parameters are bounded.…”

“…and also that c 1 > 0 and c 2 > 0, where c 1 and c 2 are both defined in (16), with C 1 and C 2 defined in (10). Then {P k } k≥1 is a nonincreasing sequence, and the following inequalities hold for any k ≥ 1,…”