Further properties of the forwardbackward envelope with applications to differenceofconvex programming
Abstract
In this paper, we further study the forwardbackward envelope first introduced in [28] and [30] for problems whose objective is the sum of a proper closed convex function and a twice continuously differentiable possibly nonconvex function with Lipschitz continuous gradient. We derive sufficient conditions on the original problem for the corresponding forwardbackward envelope to be a levelbounded and KurdykaŁojasiewicz function with an exponent of $\frac12$; these results are important for the efficient minimization of the forwardbackward envelope by classical optimization algorithms. In addition, we demonstrate how to minimize some differenceofconvex regularized least squares problems by minimizing a suitably constructed forwardbackward envelope. Our preliminary numerical results on randomly generated instances of largescale $\ell_{12}$ regularized least squares problems [37] illustrate that an implementation of this approach with a limitedmemory BFGS scheme usually outperforms standard firstorder methods such as the nonmonotone proximal gradient method in [35].
 Publication:

arXiv eprints
 Pub Date:
 April 2016
 arXiv:
 arXiv:1605.00201
 Bibcode:
 2016arXiv160500201L
 Keywords:

 Mathematics  Optimization and Control;
 Statistics  Machine Learning
 EPrint:
 Theorem 3.3 is added. Included numerical tests on oversampled DCT matrix