An improved algorithm for the $$L_2$$ L 2 – $$L_p$$ L p minimization problem

Abstract

In this paper we consider a class of non-Lipschitz and non-convex minimization problems which generalize the \(L_2\)–\(L_p\) minimization problem. We propose an iterative algorithm that decides the next iteration based on the local convexity/concavity/sparsity of its current position. We show that our algorithm finds an \(\epsilon \)-KKT point within \(O(\log {\epsilon ^{-1}})\) iterations from certain initial points. The same result is also applied to the problem with general linear constraints under mild conditions.