# Accelerated Proximal Dogleg Majorization for Sparse Regularized Quadratic Optimization Problem

## TL;DR

This paper introduces PDOME, an accelerated algorithm for sparse regularized quadratic optimization that improves convergence speed and guarantees global convergence to critical points, surpassing the original PDOM method.

## Contribution

The paper proposes PDOME, a novel accelerated algorithm with extrapolation and enhanced line search, providing global convergence analysis for nonconvex nonsmooth problems.

## Key findings

- PDOME converges faster than PDOM in numerical experiments.
- PDOME achieves better local optima in practice.
- Theoretical analysis confirms global convergence under KL property.

## Abstract

This paper addresses the problems of minimizing the sum of a quadratic function and a proximal-friendly nonconvex nonsmooth function. While the existing Proximal Dogleg Opportunistic Majorization (PDOM) algorithm for these problems offers computational efficiency by minimizing opportunistic majorization subproblems along mixed Newton directions and requiring only a single Hessian inversion, its convergence rate is limited due to the nonconvex nonsmooth regularization term, and its theoretical analysis is restricted to local convergence. To overcome these limitations, we firstly propose a novel algorithm named PDOM with extrapolation (PDOME). Its core innovations lie in two key aspects: (1) the integration of an extrapolation strategy into the construction of the hybrid Newton direction, and (2) the enhancement of the line search mechanism. Furthermore, we establish the global convergence of the entire sequence generated by PDOME to a critical point and derive its convergence rate under the Kurdyka-Lojasiewicz (KL) property. Numerical experiments demonstrate that PDOME achieves faster convergence and tends to converge to a better local optimum compared to the original PDOM.