A primal–dual penalty method via rounded weighted-ℓ₁ Lagrangian duality

Abstract

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 non-convex 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 is convex 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 non-convex 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.

Keywords:

• primal–dual methods
• penalty function methods
• ${\ell }_1$-penalty function
• Kissing number problem
• Markov–Dubins problem

Mathematical Subject Classifications:

• 49M29
• 90C26
• 90C90

Disclosure statement

No potential conflict of interest was reported by the author(s).