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18 February 2025
 
  » arxiv » 1605.0201

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Further properties of the forward-backward envelope with applications to difference-of-convex programming
Tianxiang Liu ; Ting Kei Pong ;
Date 1 May 2016
AbstractIn this paper, we further study the forward-backward envelope first introduced in [27] and [29] for problems whose objective is the sum of a proper closed convex function and a smooth possibly nonconvex function with Lipschitz continuous gradient. We derive sufficient conditions on the original problem for the corresponding forward-backward envelope to be a level-bounded and Kurdyka-{L}ojasiewicz function with an exponent of $frac12$; these results are important for the efficient minimization of the forward-backward envelope by classical optimization algorithms. In addition, we demonstrate how to minimize some difference-of-convex regularized least squares problems by minimizing a suitably constructed forward-backward envelope. Our preliminary numerical results on randomly generated instances of large-scale $ell_{1-2}$ regularized least squares problems [36] illustrate that an implementation of this approach with a limited-memory BFGS scheme outperforms some standard first-order methods such as the nonmonotone proximal gradient method in [34].
Source arXiv, 1605.0201
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