English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper, by using tools of second-order variational analysis, we studythe popular forward-backward splitting method with Beck-Teboulle's line-searchfor solving convex optimization problem where the objective function can besplit into the sum of a differentiable function and a possible nonsmoothfunction. We first establish that this method exhibits global convergence to anoptimal solution of the problem (if it exists) without the usual assumptionthat the gradient of the differentiable function involved is globally Lipschitzcontinuous. We also obtain the $o(k^{-1})$ complexity for the functional valuesequence when this usual assumption is weaken from global Lipschitz continuityto local Lipschitz continuity, improving the existing $\mathcal{O}(k^{-1})$complexity result. We then derive the local and global Q-linear convergence ofthe method in terms of both the function value sequence and the iterativesequence, under a general metric subregularity assumption which isautomatically satisfied for convex piecewise-linear-quadratic optimizationproblems. In particular, we provide verifiable sufficient conditions for metricsubregularity assumptions, and so, local and global Q-linear convergence of theproposed method for broad structured optimization problems arise in machinelearning and signal processing including Poisson linear inverse problem, thepartly smooth optimization problems, as well as the $\ell_1$-regularizedoptimization problems. Our results complement the current literature byproviding $Q$-linear convergence result to the forward-backward splittingmethod under weaker assumptions. Moreover, via this approach, we obtain severalfull characterizations for the uniqueness of optimal solution to Lasso problem,which covers some recent results in this direction.

On the Q-linear convergence of forward-backward splitting method and uniqueness of optimal solution to Lasso