Duality of optimization problems with gauge functions

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