Resumo

We propose a new duality scheme based on a sequence of smooth minorants of the weighted-ℓ1 penalty function, interpreted as a parametrized sequence of augmented Lagrangians, to solve nonconvex constrained optimization problems. For the induced sequence of dual problems, we establish strong asymptotic duality properties. Namely, we show that (i) each of the generated dual problems is a convex problem (i.e., correspond to the maximization of a concave function over a convex set) and (ii) the dual values monotonically increase to the optimal primal value. We use these properties to devise a subgradient based primal–dual method, and show that the generated primal sequence accumulates at a solution of the original problem. We illustrate the performance of the new method with three different types of test problems: A polynomial nonconvex problem, large-scale instances of the celebrated kissing number problem, and the Markov–Dubins problem. Our numerical experiments demonstrate that, when compared with the traditional implementation of a well-known smooth solver, our new method (using the same well-known solver in its subproblem) can find better quality solutions, i.e., “deeper” local minima, or solutions closer to the global minimum. Moreover, our method seems to be more time efficient, especially when the problem has a large number of constraints.

Joint work with C. Y. Kaya (University of South Australia) and C. J. Price (University of Canterbury, Christchurch, New Zealand)