  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.
