We consider an optimization problem with sub-homogeneous functions in
its objective and constraint functions. Examples of such sub-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 sub-homogeneous
optimization problems.