Policy Optimization in Robust Control: Weak Convexity and Subgradient Methods
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| Format: | Preprint |
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2025
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| _version_ | 1866915524161568768 |
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| author | Watanabe, Yuto Liao, Feng-Yi Zheng, Yang |
| author_facet | Watanabe, Yuto Liao, Feng-Yi Zheng, Yang |
| contents | Robust control seeks stabilizing policies that perform reliably under adversarial disturbances, with $\mathcal{H}_\infty$ control as a classical formulation. It is known that policy optimization of robust $\mathcal{H}_\infty$ control naturally lead to nonsmooth and nonconvex problems. This paper builds on recent advances in nonsmooth optimization to analyze discrete-time static output-feedback $\mathcal{H}_\infty$ control. We show that the $\mathcal{H}_\infty$ cost is weakly convex over any convex subset of a sublevel set. This structural property allows us to establish the first non-asymptotic deterministic convergence rate for the subgradient method under suitable assumptions. In addition, we prove a weak Polyak-Łojasiewicz (PL) inequality in the state-feedback case, implying that all stationary points are globally optimal. We finally present a few numerical examples to validate the theoretical results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_25633 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Policy Optimization in Robust Control: Weak Convexity and Subgradient Methods Watanabe, Yuto Liao, Feng-Yi Zheng, Yang Optimization and Control Systems and Control Robust control seeks stabilizing policies that perform reliably under adversarial disturbances, with $\mathcal{H}_\infty$ control as a classical formulation. It is known that policy optimization of robust $\mathcal{H}_\infty$ control naturally lead to nonsmooth and nonconvex problems. This paper builds on recent advances in nonsmooth optimization to analyze discrete-time static output-feedback $\mathcal{H}_\infty$ control. We show that the $\mathcal{H}_\infty$ cost is weakly convex over any convex subset of a sublevel set. This structural property allows us to establish the first non-asymptotic deterministic convergence rate for the subgradient method under suitable assumptions. In addition, we prove a weak Polyak-Łojasiewicz (PL) inequality in the state-feedback case, implying that all stationary points are globally optimal. We finally present a few numerical examples to validate the theoretical results. |
| title | Policy Optimization in Robust Control: Weak Convexity and Subgradient Methods |
| topic | Optimization and Control Systems and Control |
| url | https://arxiv.org/abs/2509.25633 |