Groupe d’études et de recherche en analyse des décisions

# A derivative-free trust-region augmented Lagrangian algorithm

## Charles Audet, Sébastien Le Digabel et Mathilde Peyrega

We present a new derivative-free trust-region (DFTR) algorithm to solve general nonlinear constrained problems with the use of an augmented Lagrangian method. No derivatives are used, neither for the objective function nor for the constraints. An augmented Lagrangian method, known as an effective tool to solve equality and inequality constrained optimization problems with derivatives, is exploited to minimize the subproblems, composed of quadratic models that approximate the original objective function and constraints, within a trust region. The trust region ratio which leads the classical update rules for the trust region radius is defined by comparing the true decrease of the augmented Lagrangian merit function with the expected decrease. This mechanism allows to reuse the basic unconstrained DFTR update rules with minor modifications. Computational experiments on a set of analytical problems suggest that our approach outperforms HOPSPACK and is competitive with COBYLA. Using an augmented Lagrangian, and more generally a merit function, to design the DFTR update rules with constraints is shown to be an efficient technique.

, 13 pages