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

Abstract

In a recent paper, Birgin, Floudas and Martínez introduced an augmented Lagrangian method for global optimization. In their approach, augmented Lagrangian subproblems are solved using the αBB method and convergence to global minimizers was obtained assuming feasibility of the original problem. In the present research, the algorithm mentioned above will be improved in several crucial aspects. On the one hand, feasibility of the problem will not be required. Possible infeasibility will be detected in finite time by the new algorithms and optimal infeasibility results will be proved. On the other hand, finite termination results that guarantee optimality and/or feasibility up to any required precision will be provided. An adaptive modification in which subproblem tolerances depend on current feasibility and complementarity will also be given. The adaptive algorithm allows the augmented Lagrangian subproblems to be solved without requiring unnecessary potentially high precisions in the intermediate steps of the method, which improves the overall efficiency. Experiments showing how the new algorithms and results are related to practical computations will be given.

Optimality properties of an Augmented Lagrangian method on infeasible problems

“…Thus, the gradient of this function vanishes at x * . Then, the AKKT conditions corresponding to (23) hold trivially defining all multipliers equal to zero in (17)(18)(19). Now, consider the case in which ρ k tends to infinity.…”

“…Rigorously speaking, such problems have no solutions at all and, so, a desirable property of algorithms is to detect infeasibility as soon as possible [11,19,21,22,23,30,31,46,32,39,43,47,48]. However, in many practical situations one is interested in optimizing the function, admitting some level of infeasibility.…”

“…In recent papers [19,39] we considered the problem of early detection of this fact, without concerns about the quality of the limit point obtained in terms of optimality. In the present paper we will adopt the point of view that it is relevant to distinguish between different infeasible points according to their objective function values.…”

An Augmented Lagrangian Method for Equality Constrained Optimization with Rapid Infeasibility Detection Capabilities

“…Martínez and Prudente [7] proposed an adaptive stopping criterion for the solution of the subproblems and showed that their new algorithm performs better than the original version of ALGENCAN. Birgin et al [8,9] improved also an augmented Lagrangian algorithm within the framework of global optimization and show better performances than the initial implementation in [10]. Gonçalves et al [11] extend the results of [8] to an entire class of penalty functions.…”

“…Birgin et al [8,9] improved also an augmented Lagrangian algorithm within the framework of global optimization and show better performances than the initial implementation in [10]. Gonçalves et al [11] extend the results of [8] to an entire class of penalty functions. Within the framework of sequential quadratic programming (SQP) methods, Byrd et al [12] have proposed an algorithm to quickly detect infeasibility and have shown that their algorithm has fast local convergence properties.…”