Quadratic Growth Conditions and Uniqueness of Optimal Solution to Lasso

Abstract

In the previous paper Bello-Cruz et al. (J Optim Theory Appl 188:378–401, 2021), we showed that the quadratic growth condition plays a key role in obtaining Q-linear convergence of the widely used forward–backward splitting method with Beck–Teboulle’s line search. In this paper, we analyze the property of quadratic growth condition via second-order variational analysis for various structured optimization problems that arise in machine learning and signal processing. This includes, for example, the Poisson linear inverse problem as well as the ell _1-regularized optimization problems. As a by-product of this approach, we also obtain several full characterizations for the uniqueness of optimal solution to Lasso problem, which complements and extends recent important results in this direction.