Bundle methods for dual atomic pursuit

TL;DR

This paper introduces a two-stage bundle method algorithm leveraging gauge duality for structured optimization, efficiently identifying atomic support and recovering primal solutions in large-scale problems.

Contribution

It proposes a novel two-stage approach combining gauge duality and bundle methods for atomic pursuit in structured optimization.

Findings

Efficient algorithms for large-scale structured optimization.

Effective atomic support identification via dual problem approximations.

Primal solutions recovered accurately from dual support.

Abstract

The aim of structured optimization is to assemble a solution, using a given set of (possibly uncountably infinite) atoms, to fit a model to data. A two-stage algorithm based on gauge duality and bundle method is proposed. The first stage discovers the optimal atomic support for the primal problem by solving a sequence of approximations of the dual problem using a bundle-type method. The second stage recovers the approximate primal solution using the atoms discovered in the first stage. The overall approach leads to implementable and efficient algorithms for large problems.