Weak Convexity and Proximal Bundle Methods for Nonsmooth Policy Optimization in Robust Control
Abstract
We study policy optimization for discrete-time robust ℋ∞ control with static output-feedback, and present the first feasibility-preserving algorithm with a deterministic, non-asymptotic complexity guarantee. This problem naturally leads to a nonsmooth and nonconvex optimization over the set of stabilizing feedback gains. We first establish several structural properties of the ℋ∞ cost. In particular, we show that the cost is weakly convex on every convex subset of a sublevel set. For the state-feedback case, we further establish a weak Polyak--Łojasiewicz inequality, which ensures that every stationary point is globally optimal. Building on these properties, we develop a proximal bundle method for ℋ∞ policy optimization. The proposed method can be viewed as an implementable approximation of the proximal point method and uses only function value and subgradient information. We show that all iterates remain stabilizing and establish a deterministic non-asymptotic complexity bound of 𝒪(max{η−4,ε−2}) for finding an (η,ε)-stationary point. Numerical experiments illustrate our theoretical results.
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BibTeX
@misc{watanabe2026weak,
title = {{Weak Convexity and Proximal Bundle Methods for Nonsmooth Policy Optimization in Robust Control}},
author = {Yuto Watanabe and Feng-Yi Liao and Yang Zheng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06202},
eprint = {2609.06202},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06202},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Weak Convexity and Proximal Bundle Methods for Nonsmooth Policy Optimization in Robust Control
AU - Yuto Watanabe
AU - Feng-Yi Liao
AU - Yang Zheng
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.06202
UR - https://arxiv.org/abs/2609.06202
AB - We study policy optimization for discrete-time robust ℋ_∞ control with static output-feedback, and present the first feasibility-preserving algorithm with a deterministic, non-asymptotic complexity guarantee. This problem naturally leads to a nonsmooth and nonconvex optimization over the set of stabilizing feedback gains. We first establish several structural properties of the ℋ_∞ cost. In particular, we show that the cost is weakly convex on every convex subset of a sublevel set. For the state-feedback case, we further establish a weak Polyak--Łojasiewicz inequality, which ensures that every stationary point is globally optimal. Building on these properties, we develop a proximal bundle method for ℋ_∞ policy optimization. The proposed method can be viewed as an implementable approximation of the proximal point method and uses only function value and subgradient information. We show that all iterates remain stabilizing and establish a deterministic non-asymptotic complexity bound of 𝒪(max{η^−4,ε^−2}) for finding an (η,ε)-stationary point. Numerical experiments illustrate our theoretical results.
ER -