Algorithms for Difference-of-Convex (DC) Programs Based on Difference-of-Moreau-Envelopes Smoothing

Abstract

In this paper we consider minimization of a difference-of-convex (DC) function with and without linear constraints. We first study a smooth approximation of a generic DC function, termed difference-of-Moreau-envelopes (DME) smoothing, where both components of the DC function are replaced by their respective Moreau envelopes. The resulting smooth approximation is shown to be Lipschitz differentiable, capture stationary points, local, and global minima of the original DC function, and enjoy some growth conditions, such as level-boundedness and coercivity, for broad classes of DC functions. We then develop four algorithms for solving DC programs with and without linear constraints based on the DME smoothing. In particular, for a smoothed DC program without linear constraints, we show that the classic gradient descent method as well as an inexact variant can obtain a stationary solution in the limit with a convergence rate of $\mathcal{O}(K^{-1/2})$, where $K$ is the number of proximal evaluations of both components. Furthermore, when the DC program is explicitly constrained in an affine subspace, we combine the smoothing technique with the augmented Lagrangian function and derive two variants of the augmented Lagrangian method (ALM), named LCDC-ALM and composite LCDC-ALM, focusing on different structures of the DC objective function. We show that both algorithms find an $\epsilon$-approximate stationary solution of the original DC program in $\mathcal{O}({\epsilon }^{-2})$ iterations. Comparing to existing methods designed for linearly constrained weakly convex minimization, the proposed ALM-based algorithms can be applied to a broader class of problems, where the objective contains a nonsmooth concave component. Finally, numerical experiments are presented to demonstrate the performance of the proposed algorithms.

References in corpus (8)

• Stochastic subgradient method converges at the rate $O(k^{-1/4})$ on weakly convex functions
• The proximal point method revisited
• Decomposing Linearly Constrained Nonconvex Problems by a Proximal Primal Dual Approach: Algorithms, Convergence, and Applications
• An Augmented Lagrangian Method for Conic Convex Programming
• Iteration complexity of first-order augmented Lagrangian methods for convex conic programming
• Iteration-complexity of an inexact proximal accelerated augmented Lagrangian method for solving linearly constrained smooth nonconvex composite optimization problems
• A global dual error bound and its application to the analysis of linearly constrained nonconvex optimization
• Augmented Lagrangian based first-order methods for convex-constrained programs with weakly-convex objective

Cited by in corpus (1)

• Dual Descent ALM and ADMM