Abstract
In this paper, we consider nonlinear optimization problems with nonlinear equality constraints and bound constraints on the variables. For the solution of such problems, many augmented Lagrangian methods have been defined in the literature. Here, we propose to modify one of these algorithms, namely ALGENCAN by Andreani et al., in such a way to incorporate second-order information into the augmented Lagrangian framework, using an active-set strategy. We show that the overall algorithm has the same convergence properties as ALGENCAN and an asymptotic quadratic convergence rate under suitable assumptions. The numerical results confirm that the proposed algorithm is a viable alternative to ALGENCAN with greater robustness.
Highlights
In this paper, we are interested in the solution of smooth constrained optimization problems of the type: min f (x) h(x) = 0 (1)≤ x ≤ u, where x, u ∈ n, i < ui, for all i = 1, . . . , n, f : n →, h : n → p are twice continuously differentiable functions
The algorithm we propose here builds upon the augmented Lagrangian method described in [3], where an augmented Lagrangian function is defined with respect to a subset of constraints and iteratively minimized over x subject to the remaining constraints
The result stated in the above proposition holds for an unknown neighborhood of the optimal solution. It would be of great interest and importance to give a characterization of that neighborhood, in order to bound the maximum number of iterations required by the algorithm to identify the active set. This is an open problem and we think it may represent a possible line of future research, for example by adapting the complexity results given for ALGENCAN in [7], or extending some results on finite active-set identification given in the literature for specific classes of algorithms [8,10,22]
Summary
The computational heavy part of ALGENCAN consists in the solution (at every outer iteration) of the subproblem, i.e., the minimization of the augmented Lagrangian merit function for given values of the penalty parameter and of the estimated Lagrange multipliers. Such minimization is carried out by the inner solver GENCAN [5]. On the other hand, when the Newton-type direction cannot be computed or does not satisfy a proper condition, we propose to resort to the minimization of the augmented Lagrangian function, but using an efficient active-set method for bound-constrained problems [11].
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