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Generic linear convergence through metric subregularity in a variable-metric extension of the proximal point algorithm

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Abstract

The proximal point algorithm finds a zero of a maximal monotone mapping by iterations in which the mapping is made strongly monotone by the addition of a proximal term. Here it is articulated with the norm behind the proximal term possibly shifting from one iteration to the next, but under conditions that eventually make the metric settle down. Despite the varying geometry, the sequence generated by the algorithm is shown to converge to a particular solution. Although this is not the first variable-metric extension of proximal point algorithm, it is the first to retain the flexibility needed for applications to augmented Lagrangian methodology and progressive decoupling. Moreover, in a generic sense, the convergence it generates is Q-linear at a rate that depends in a simple way on the modulus of metric subregularity of the mapping at that solution. This is a tighter rate than previously identified and reveals for the first time the definitive role of metric subregularity in how the proximal point algorithm performs, even in fixed-metric mode.

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  • 06 November 2023

    The math code is not rendered properly and it has been corrected.

Notes

  1. This rate is given by the value \(\nu \) at the top of page 254 of [4], which is indicated as depending on a value \(\mu \) and a stepsize relaxation parameter \(\theta \in (0,1)\). But \(\mu =\sqrt{\alpha ^2+1}\sqrt{\beta ^2-1}+\alpha \beta \) where \(\alpha \) already depends on \(\theta \) and both \(\alpha \) and \(\beta \) depend also on another parameter \({\bar{\sigma }}\).

  2. The proof says this follows from Fejér monotonicity, but offers no evidence for Fejér monotonicity, which is indeed doubtful for the situation at hand.

  3. This paper is dedicated to the memory of Asen Dontchev, who died 16 September 2021. Metric regularity was a subject very dear to him.

  4. By avoiding a ratio, this accommodates the possibility that \(||z^k-{\bar{z}}||\) might sometimes be 0 but not stay at 0.

  5. For the set R that is the closure of the range of T, this being convex because of T being maximal monotone, the normal cone at 0 must contain the normal cones at all points of R in some neighborhood of 0, as holds for instance when R is polyhedral.

  6. With their parameters \(\theta =0\) and \(\varepsilon _k=0\); our \(B_k\) corresponds to their \(M_k^{-1}\). However they actually require \(\theta \in (0,1)\)!

  7. Plenty of criteria are available for this; see for instance [16, 12.51]. Classically it’s known that emptiness is signaled by PPA iterates “converging to the horizon,” but this isn’t taken up here for the current algorithm.

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  1. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195-4350, USA

    R. Tyrrell Rockafellar

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Rockafellar, R.T. Generic linear convergence through metric subregularity in a variable-metric extension of the proximal point algorithm. Comput Optim Appl 86, 1327–1346 (2023). https://doi.org/10.1007/s10589-023-00494-z

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