Strong Convergence of FISTA for Affinely Constrained Convex Quadratic Minimization

TL;DR

This paper proves strong convergence of FISTA and Nesterov's accelerated gradient method for convex quadratic minimization, removing previous assumptions and resolving longstanding open problems in optimization.

Contribution

It establishes strong convergence results for FISTA in affinely constrained convex quadratic problems without closedness conditions, extending to unconstrained cases.

Findings

FISTA converges strongly for affinely constrained convex quadratic problems.

Nesterov's accelerated gradient method converges strongly for convex quadratic objectives.

The results remove previous assumptions required for convergence proofs.

Abstract

In October 2025, research by Bo\c{t}, Fadili, and Nguyen, and by Jang and Ryu, led to the seminal result that Beck and Teboulle's FISTA converges weakly to a minimizer of the sum of two convex functions resolving a long-standing open problem. The first strong convergence result was obtained in November 2025 by Moursi, Naguib, Pavlovic, and Vavasis for affinely constrained convex minimization provided certain closedness conditions hold. In this paper, we prove strong convergence in the affine-quadratic case without any closedness assumption. Specializing this to the unconstrained case, we obtain the strong convergence of Nesterov's accelerated gradient method when applied to a convex quadratic objective function.