Semiglobal exponential stability of the discrete-time Arrow-Hurwicz-Uzawa primal-dual algorithm for constrained optimization

We consider the discrete-time Arrow-Hurwicz-Uzawa primal-dual algorithm, also known as the first-order Lagrangian method, for constrained optimization problems involving a smooth strongly convex cost and smooth convex constraints. We prove that, for every given compact set of initial conditions, the...

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Bibliographic Details
Published in:Mathematical programming Vol. 208; no. 1-2; pp. 629 - 660
Main Authors: Bin, Michelangelo, Notarnicola, Ivano, Parisini, Thomas
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2024
Springer
Subjects:
Algorithms
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Theoretical
68Q25
93C55
Primal-dual algorithm
90C25
Constraint programming
Discrete-time control/observation systems
Convex programming
37N40
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:We consider the discrete-time Arrow-Hurwicz-Uzawa primal-dual algorithm, also known as the first-order Lagrangian method, for constrained optimization problems involving a smooth strongly convex cost and smooth convex constraints. We prove that, for every given compact set of initial conditions, there always exists a sufficiently small stepsize guaranteeing exponential stability of the optimal primal-dual solution of the problem with a domain of attraction including the initialization set. Inspired by the analysis of nonlinear oscillators, the stability proof is based on a non-quadratic Lyapunov function including a nonlinear cross term.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-023-02051-2