Automated tight Lyapunov analysis for first-order methods

We present a methodology for establishing the existence of quadratic Lyapunov inequalities for a wide range of first-order methods used to solve convex optimization problems. In particular, we consider (i) classes of optimization problems of finite-sum form with (possibly strongly) convex and possib...

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Vydáno v:Mathematical programming
Hlavní autoři: Upadhyaya, Manu, Banert, Sebastian, Taylor, Adrien B, Giselsson, Pontus
Médium: Paper Journal Article
Jazyk:angličtina
Vydáno: Ithaca Cornell University Library, arXiv.org 22.10.2025
Springer Verlag
Témata:
Convergence
Convexity
Inequalities
Linear operators
Mathematics
Methodology
Optimization
Optimization and Control
Optimization and Control (math.OC)
FOS: Mathematics
ISSN:0025-5610, 2331-8422, 1436-4646
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Popis
Shrnutí:We present a methodology for establishing the existence of quadratic Lyapunov inequalities for a wide range of first-order methods used to solve convex optimization problems. In particular, we consider (i) classes of optimization problems of finite-sum form with (possibly strongly) convex and possibly smooth functional components, (ii) first-order methods that can be written as a linear system on state-space form in feedback interconnection with the subdifferentials of the functional components of the objective function, and (iii) quadratic Lyapunov inequalities that can be used to draw convergence conclusions. We present a necessary and sufficient condition for the existence of a quadratic Lyapunov inequality within a predefined class of Lyapunov inequalities, which amounts to solving a small-sized semidefinite program. We showcase our methodology on several first-order methods that fit the framework. Most notably, our methodology allows us to significantly extend the region of parameter choices that allow for duality gap convergence in the Chambolle-Pock method when the linear operator is the identity mapping.
Bibliografie:SourceType-Working Papers-1
ObjectType-Working Paper/Pre-Print-1
content type line 50
ISSN:0025-5610
2331-8422
1436-4646
DOI:10.48550/arxiv.2302.06713