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.

我们提出了一种新的基于加权的平滑次要序列的对偶方案${\ell }_{\mathrm{1个}}$惩罚函数，解释为增广拉格朗日函数的参数化序列，用于解决非凸约束优化问题。对于对偶问题的导出序列，我们建立了强渐近对偶性。即，我们表明 (i) 对偶问题的序列是凸的，并且 (ii) 对偶值单调增加到最优原始值。我们使用这些属性来设计基于次梯度的原始对偶方法，并表明生成的原始序列在原始问题的解中累积。我们用三种不同类型的测试问题来说明新方法的性能：多项式非凸问题、著名的接吻数问题的大规模实例以及 Markov–Dubins 问题。我们的数值实验表明，与众所周知的平滑求解器的传统实现相比，我们的新方法（在其子问题中使用相同的著名求解器）可以找到质量更好的解决方案，即“更深”的局部最小值，或更接近全局最小值的解决方案。此外，我们的方法似乎更省时，尤其是当问题具有大量约束时。