Duality of optimization problems with gauge functions

Shota Yamanaka, Nobuo Yamashita

Published 2017-12-13Version 1

Recently, Yamanaka and Yamashita (2017) proposed the so-called positively homogeneous optimization problems, which generalize many important problems, in particular the absolute-value and the gauge optimizations. They presented a closed dual formulation for these problems, proving weak duality results, and showing that it is equivalent to the Lagrangian dual under some conditions. In this work, we focus particularly in optimization problems whose objective functions and constraints consist of some gauge and linear functions. Through the positively homogeneous framework, we prove that both weak and strong duality results hold. We also discuss necessary and sufficient optimality conditions associated to these problems. Finally, we show that it is possible to recover primal solutions from Karush-Kuhn-Tucker points of the dual formulation.