Iterative methods for projection onto the ℓ p quasi-norm ball

Abstract

In this paper, we focus on computing the projection onto the ℓ p quasi-norm ball. Motivated by the existing IRBP algorithm, which tackles ℓ p quasi-norm ball projection through projections onto a sequence of simple and tractable weighted ℓ 1 norm balls, we propose an innovative and effective iteratively reweighting approach for solving the projection problem. Specifically, our method centres on an ϵ -approximation that yields better approximation accuracy. We propose an innovative non-smooth, yet continuously differentiable approximation to the ℓ p quasi-norm function. Leveraging the concavity of our approximate model, we developed a novel variant of the iterative reweighted ℓ 1 norm ball projection algorithm. Through rigorous analysis, we demonstrated the global convergence properties of our proposed numerical approach. Numerical studies demonstrates the superior computational efficiency of our proposed algorithm. Moreover, our solution method can be further leveraged within current iterative algorithms designed to solve ℓ p quasi-norm ball constrained optimization problems, especially in scenarios where rapid convergence is of utmost importance.