On a family of relaxed gradient descent methods for strictly convex quadratic minimization

Liam MacDonald (), Rua Murray () and Rachael Tappenden ()
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Liam MacDonald: University of Canterbury
Rua Murray: University of Canterbury
Rachael Tappenden: University of Canterbury

Computational Optimization and Applications, 2025, vol. 91, issue 1, No 6, 173-200

Abstract: Abstract This paper studies the convergence properties of a family of Relaxed $$\ell $$ ℓ -Minimal Gradient Descent methods for quadratic optimization; the family includes the omnipresent Steepest Descent method, as well as the Minimal Gradient method. Simple proofs are provided that show, in an appropriately chosen norm, the gradient and the distance of the iterates from optimality converge linearly, for all members of the family. Moreover, the function values decrease linearly, and iteration complexity results are provided. All theoretical results hold when (fixed) relaxation is employed. It is also shown that, given a fixed overhead and storage budget, every Relaxed $$\ell $$ ℓ -Minimal Gradient Descent method can be implemented using exactly one matrix vector product. Numerical experiments are presented that illustrate the benefits of relaxation across the family.

Keywords: Steepest descent; Relaxation; Linear convergence; Quadratic optimization; Strong convexity; Positive definite Hessian (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10589-025-00670-3

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