KL property of exponent $1/2$ of quadratic functions under nonnegative zero-norm constraints and applications

TL;DR

This paper proves the KL property of exponent 1/2 for certain nonconvex, nonsmooth quadratic optimization problems with nonnegative zero-norm constraints and develops a globally convergent PGD method, validated by numerical experiments.

Contribution

It establishes the KL property of exponent 1/2 for specific nonconvex problems and introduces a globally convergent projection gradient descent method.

Findings

KL property of exponent 1/2 proven for the objective functions

Proposed PGD method achieves global and linear convergence

Numerical experiments confirm theoretical results on real and synthetic data

Abstract

This paper focuses on the quadratic optimization over two classes of nonnegative zero-norm constraints: nonnegative zero-norm sphere constraint and zero-norm simplex constraint, which have important applications in nonnegative sparse eigenvalue problems and sparse portfolio problems, respectively. We establish the KL property of exponent 1/2 for the extended-valued objective function of these nonconvex and nonsmooth optimization problems, and use this crucial property to develop a globally and linearly convergent projection gradient descent (PGD) method. Numerical results are included for nonegative sparse principal component analysis and sparse portfolio problems with synthetic and real data to confirm the theoretical results.