A Deflected Subgradient Method Using a General Augmented Lagrangian Duality with Implications on Penalty Methods

We propose a duality scheme for solving constrained nonsmooth and nonconvex optimization problems. Our approach is to use a new variant of the deflected subgradient method for solving the dual problem. Our augmented Lagrangian function induces a primal–dual method with strong duality, that is, with zero duality gap. We prove that our method converges to a dual solution if and only if a dual solution exists. We also prove that all accumulation points of an auxiliary primal sequence are primal solutions. Our results apply, in particular, to classical penalty methods, since the penalty functions associated with these methods can be recovered as a special case of our augmented Lagrangians. Besides the classical augmenting terms given by the ℓ 1- or ℓ 2-norm forms, terms of many other forms can be used in our Lagrangian function. Using a practical selection of the step-size parameters, as well as various choices of the augmenting term, we demonstrate the method on test problems. Our numerical experiments indicate that it is more favourable to use an augmenting term of an exponential form rather than the classical ℓ 1- or ℓ 2-norm forms.