Constrained Optimization Involving Nonconvex $\ell_p$ Norms: Optimality Conditions, Algorithm and Convergence

TL;DR

This paper develops optimality conditions and convergence analysis for constrained optimization problems involving the nonconvex _p norm with 0<p<1, which are important for promoting sparsity but are challenging due to nonsmoothness.

Contribution

It provides the calculation of subgradients, normal cones, and first-order necessary conditions for _p norm problems, along with convergence analysis of iterative algorithms.

Findings

Derived subgradients and normal cones for _p norms.

Established first-order necessary conditions under various constraints.

Demonstrated global convergence of reweighted algorithms.

Abstract

This paper investigates the optimality conditions for characterizing the local minimizers of the constrained optimization problems involving an $\ell_p$ norm ($0<p<1$) of the variables, which may appear in either the objective or the constraint. This kind of problems have strong applicability to a wide range of areas since usually the $\ell_p$ norm can promote sparse solutions. However, the nonsmooth and non-Lipschtiz nature of the $\ell_p$ norm often cause these problems difficult to analyze and solve. We provide the calculation of the subgradients of the $\ell_p$ norm and the normal cones of the $\ell_p$ ball. For both problems, we derive the first-order necessary conditions under various constraint qualifications. We also derive the sequential optimality conditions for both problems and study the conditions under which these conditions imply the first-order necessary conditions. We point out that the sequential optimality conditions can be easily satisfied for iteratively reweighted algorithms and show that the global convergence can be easily derived using sequential optimality conditions.