  We consider an optimization problem with positively homogeneous functions in
its objective and constraint functions. Examples of such positively homogeneous
functions include the absolute value function and the $p$-norm function, where
$p$ is a positive real number. The problem, which is not necessarily convex,
extends the absolute value optimization proposed in [O. L. Mangasarian,
Absolute value programming, Computational Optimization and Applications 36
(2007) pp. 43-53]. In this work, we propose a dual formulation that,
differently from the Lagrangian dual approach, has a closed-form and some
interesting properties. In particular, we discuss the relation between the
Lagrangian duality and the one proposed here, and give some sufficient
conditions under which these dual problems coincide. Finally, we show that some
well-known problems, e.g., sum of norms optimization and the group Lasso-type
optimization problems, can be reformulated as positively homogeneous
optimization problems.
