Accelerating Level-Value Adjustment for the Polyak Stepsize Accelerating Level-Value Adjustment for the Polyak Stepsize

The Polyak stepsize has been widely used in subgradient methods for non-smooth convex optimization. However, calculating the stepsize requires the optimal value, which is generally unknown. Therefore, dynamic estimations of the optimal value are usually needed. In this paper, to guarantee convergenc...

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Vydáno v:Journal of optimization theory and applications Ročník 206; číslo 3; s. 71
Hlavní autoři: Liu, Anbang, Bragin, Mikhail A., Chen, Xi, Guan, Xiaohong
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.09.2025
Springer Nature B.V
Témata:
Applications of Mathematics
Calculus of Variations and Optimal Control; Optimization
Convergence
Convex analysis
Convexity
Engineering
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Sensors
Theory of Computation
Approximate subgradient method
Polyak stepsize
Convex optimization
90C25
Non-smooth optimization
Subgradient method
ISSN:0022-3239, 1573-2878
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Popis
Shrnutí:The Polyak stepsize has been widely used in subgradient methods for non-smooth convex optimization. However, calculating the stepsize requires the optimal value, which is generally unknown. Therefore, dynamic estimations of the optimal value are usually needed. In this paper, to guarantee convergence, a series of level values is constructed to estimate the optimal value successively. This is achieved by developing a decision-guided procedure that involves solving a novel, easy-to-solve linear constraint satisfaction problem referred to as the “Polyak Stepsize Violation Detector” (PSVD). Once a violation is detected, the level value is recalculated. We rigorously establish the convergence for both the level values and the objective function values. Furthermore, with our level adjustment approach, calculating an approximate subgradient in each iteration is sufficient for convergence. A series of empirical tests of convex optimization problems with diverse characteristics demonstrates the practical advantages of our approach over existing methods.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-025-02750-0