# Restart FISTA with Global Linear Convergence

## TL;DR

This paper introduces a new restart scheme for FISTA that guarantees global linear convergence in non-strongly convex problems satisfying quadratic growth, without needing prior knowledge of certain parameters.

## Contribution

The paper proposes a novel restart scheme for FISTA that achieves global linear convergence without requiring prior parameter knowledge.

## Key findings

- Outperforms existing restart FISTA schemes in simulations
- Achieves global linear convergence for non-strongly convex problems
- Does not require prior knowledge of the objective's optimal value or growth parameter

## Abstract

Fast Iterative Shrinking-Threshold Algorithm (FISTA) is a popular fast gradient descent method (FGM) in the field of large scale convex optimization problems. However, it can exhibit undesirable periodic oscillatory behaviour in some applications that slows its convergence. Restart schemes seek to improve the convergence of FGM algorithms by suppressing the oscillatory behaviour. Recently, a restart scheme for FGM has been proposed that provides linear convergence for non strongly convex optimization problems that satisfy a quadratic functional growth condition. However, the proposed algorithm requires prior knowledge of the optimal value of the objective function or of the quadratic functional growth parameter. In this paper we present a restart scheme for FISTA algorithm, with global linear convergence, for non strongly convex optimization problems that satisfy the quadratic growth condition without requiring the aforementioned values. We present some numerical simulations that suggest that the proposed approach outperforms other restart FISTA schemes.