Duality of nonconvex optimization with positively homogeneous functions

TL;DR

This paper introduces a novel dual formulation for optimization problems involving positively homogeneous functions, providing a closed-form solution and exploring its relation to traditional Lagrangian duality, with applications to norm-based problems.

Contribution

It proposes a new dual approach for nonconvex positively homogeneous optimization problems, differing from Lagrangian duals, with conditions for equivalence and practical reformulations.

Findings

The dual formulation has a closed-form expression.

Conditions under which dual problems coincide are identified.

Applications include sum of norms and group Lasso optimization problems.

Abstract

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.