Global minimization using an Augmented Lagrangian method with variable lower-level constraints

Abstract

A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration k the method requires the ε k -global minimization of the Augmented Lagrangian with simple constraints, where ε k → ε. Global convergence to an ε-global minimizer of the original problem is proved. The subproblems are solved using the αBB method. Numerical experiments are presented.

Augmented Lagrangians with possible infeasibility and finite termination for global nonlinear programming

“…that a problem with difficult-to-handle constraints may be tackled by solving a sequence of subproblems with simpler constraints using some well-established existing method. In [21] and the present paper we exploit the ability of αBB to solve linearly constrained global optimization problems, which has been corroborated in many applied papers. In order to take advantage of the αBB potentialities, augmented Lagrangian subproblems are "over-restricted" by means of linear constraints that simplify subproblem resolutions and do not affect a successful search of global minimizers.…”

“…The first coincides essentially with the one introduced in [21] and solves each subproblem with a precision ε k that tends to zero. In the second version we employ an adaptive precision control that depends on the infeasibility of iterates of internal iterations.…”

“…In simple constrained cases, several well-known algorithms accomplish that purpose efficiently. This is the case of the αBB algorithm [1,2,3,14], that has been used in [21] as subproblems solver in the context of an augmented Lagrangian method.…”

“…The algorithm introduced in [21] for constrained global optimization was based on the PowellHestenes-Rockafellar (PHR) augmented Lagrangian approach [46,59,61]. An implementation in which subproblems were solved by means of the αBB method was described and tested in [21].…”

“…An implementation in which subproblems were solved by means of the αBB method was described and tested in [21]. The convergence theory of [21] assumes that the nonlinear programming problem is feasible and proves that limit points of sequences generated by the algorithm are ε-global minimizers, where ε is a given positive tolerance. However, a test for verifying ε-optimality at each iterate x k was not provided.…”