A successive difference-of-convex approximation method for a class of nonconvex nonsmooth optimization problems

by   Tianxiang Liu, et al.

We consider a class of nonconvex nonsmooth optimization problems whose objective is the sum of a nonnegative smooth function and a bunch of nonnegative proper closed possibly nonsmooth functions (whose proximal mappings are easy to compute), some of which are further composed with linear maps. This kind of problems arises naturally in various applications when different regularizers are introduced for inducing simultaneous structures in the solutions. Solving these problems, however, can be challenging because of the coupled nonsmooth functions: the corresponding proximal mapping can be hard to compute so that standard first-order methods such as the proximal gradient algorithm cannot be applied efficiently. In this paper, we propose a successive difference-of-convex approximation method for solving this kind of problems. In this algorithm, we approximate the nonsmooth functions by their Moreau envelopes in each iteration. Making use of the simple observation that Moreau envelopes of nonnegative proper closed functions are continuous difference-of-convex functions, we can then approximately minimize the approximation function by first-order methods with suitable majorization techniques. These first-order methods can be implemented efficiently thanks to the fact that the proximal mapping of each nonsmooth function is easy to compute. Under suitable assumptions, we prove that the sequence generated by our method is bounded and clusters at a stationary point of the objective. We also discuss how our method can be applied to concrete applications such as nonconvex fused regularized optimization problems and simultaneously structured matrix optimization problems, and illustrate the performance numerically for these two specific applications.


page 1

page 2

page 3

page 4


A block inertial Bregman proximal algorithm for nonsmooth nonconvex problems

In this paper, a block inertial Bregman proximal algorithm, namely [], f...

A Proximal Zeroth-Order Algorithm for Nonconvex Nonsmooth Problems

In this paper, we focus on solving an important class of nonconvex optim...

Stochastic Difference-of-Convex Algorithms for Solving nonconvex optimization problems

The paper deals with stochastic difference-of-convex functions programs,...

Block Alternating Bregman Majorization Minimization with Extrapolation

In this paper, we consider a class of nonsmooth nonconvex optimization p...

Sparse Optimization Problem with s-difference Regularization

In this paper, a s-difference type regularization for sparse recovery pr...

Iteratively reweighted ℓ_1 algorithms with extrapolation

Iteratively reweighted ℓ_1 algorithm is a popular algorithm for solving ...

Successive Convex Approximation Algorithms for Sparse Signal Estimation with Nonconvex Regularizations

In this paper, we propose a successive convex approximation framework fo...

Please sign up or login with your details

Forgot password? Click here to reset