An adaptive proximal safeguarded augmented Lagrangian method for nonsmooth DC problems with convex constraints
Christian Kanzow, Tanja Neder
TL;DR
This paper addresses constrained nonsmooth DC optimization by minimizing $f(x)=g(x)-h(x)$ subject to $Ax=b$, $c(x)\le0$, and $x\in C$, where both DC components may be nonsmooth. It introduces a proximal safeguarded augmented Lagrangian method (psALMDC) that linearizes the concave part $-h$ to produce convex subproblems and embeds them in an augmented Lagrangian framework with a proximal term for stability. A key theoretical contribution is showing subsequential convergence of the primal–dual sequence to a generalized KKT point under a modified Slater-type constraint qualification (mSCQ); further, when the proximal/penalty parameters grow, convergence to a generalized KKT point is established under the same qualification. Numerically, psALMDC is compared against DC algorithms (DCA) and proximal bundle methods (PBMDC) on location planning and sparse recovery problems, where it often achieves higher success rates and competitive runtimes, illustrating practicality for nonsmooth constrained DC problems with convex constraints.
Abstract
A proximal safeguarded augmented Lagrangian method for minimizing the difference of convex (DC) functions over a nonempty, closed and convex set with additional linear equality as well as convex inequality constraints is presented. Thereby, all functions involved may be nonsmooth. Iterates (of the primal variable) are obtained by solving convex optimization problems as the concave part of the objective function gets approximated by an affine linearization. Under the assumption of a modified Slater constraint qualification, both convergence of the primal and dual variables to a generalized Karush-Kuhn-Tucker (KKT) point is proven, at least on a subsequence. Numerical experiments and comparison with existing solution methods are presented using some classes of constrained and nonsmooth DC problems.