A Primal–Dual Penalty Method via Rounded Weighted-\boldmath{$\ell_1$} Lagrangian Duality

We propose a new duality scheme based on a sequence of smooth minorants of the weighted-$\ell_1$ penalty function, interpreted as a parametrized sequence of augmented Lagrangians, to solve nonconvex and nonsmooth constrained optimization problems. For the induced sequence of dual problems, we establish strong asymptotic duality properties. Namely, we show that (i) the sequence of dual problems are convex and (ii) the dual values monotonically increase converging 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 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 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.



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