Duality of optimization problems with gauge functions

TL;DR

This paper investigates the duality properties of a specific class of positively homogeneous optimization problems involving gauge and linear functions, establishing strong duality, optimality conditions, and solution recovery methods.

Contribution

It extends previous work by proving strong duality and deriving optimality conditions for gauge-based optimization problems, and discusses extensions to general convex problems.

Findings

Proved strong duality for a class of gauge and linear function optimization problems.

Derived necessary and sufficient optimality conditions.

Provided conditions for primal solution recovery from dual solutions.

Abstract

Recently, Yamanaka and Yamashita proposed the so-called positively homogeneous optimization problem, which includes many important problems, such as the absolute-value and the gauge optimizations. They presented a closed form of the dual formulation for the problem, and showed weak duality and the equivalence to the Lagrangian dual under some conditions. In this work, we focus on a special positively homogeneous optimization problem, whose objective function and constraints consist of some gauge and linear functions. We prove not only weak duality but also strong duality. We also study necessary and sufficient optimality conditions associated to the problem. Moreover, we give sufficient conditions under which we can recover a primal solution from a Karush-Kuhn-Tucker point of the dual formulation. Finally, we discuss how to extend the above results to general convex optimization problems by considering the so-called perspective functions.