Time-Varying Convex Optimization: A Contraction and Equilibrium Tracking Approach
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866915405186990080 |
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| author | Davydov, Alexander Centorrino, Veronica Gokhale, Anand Russo, Giovanni Bullo, Francesco |
| author_facet | Davydov, Alexander Centorrino, Veronica Gokhale, Anand Russo, Giovanni Bullo, Francesco |
| contents | In this article, we provide a novel and broadly-applicable contraction-theoretic approach to continuous-time time-varying convex optimization. For any parameter-dependent contracting dynamics, we show that the tracking error is asymptotically proportional to the rate of change of the parameter and that the proportionality constant is upper bounded by Lipschitz constant in which the parameter appears divided by the contraction rate of the dynamics squared. We additionally establish that augmenting any parameter-dependent contracting dynamics with a feedforward prediction term ensures that the tracking error vanishes exponentially quickly. To apply these results to time-varying convex optimization, we establish the strong infinitesimal contractivity of dynamics solving three canonical problems: monotone inclusions, linear equality-constrained problems, and composite minimization problems. For each case, we derive the sharpest-known contraction rates and provide explicit bounds on the tracking error between solution trajectories and minimizing trajectories. We validate our theoretical results on two numerical examples and on an application to control barrier function-based controller design that involves real hardware. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_15595 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Time-Varying Convex Optimization: A Contraction and Equilibrium Tracking Approach Davydov, Alexander Centorrino, Veronica Gokhale, Anand Russo, Giovanni Bullo, Francesco Optimization and Control Systems and Control Signal Processing In this article, we provide a novel and broadly-applicable contraction-theoretic approach to continuous-time time-varying convex optimization. For any parameter-dependent contracting dynamics, we show that the tracking error is asymptotically proportional to the rate of change of the parameter and that the proportionality constant is upper bounded by Lipschitz constant in which the parameter appears divided by the contraction rate of the dynamics squared. We additionally establish that augmenting any parameter-dependent contracting dynamics with a feedforward prediction term ensures that the tracking error vanishes exponentially quickly. To apply these results to time-varying convex optimization, we establish the strong infinitesimal contractivity of dynamics solving three canonical problems: monotone inclusions, linear equality-constrained problems, and composite minimization problems. For each case, we derive the sharpest-known contraction rates and provide explicit bounds on the tracking error between solution trajectories and minimizing trajectories. We validate our theoretical results on two numerical examples and on an application to control barrier function-based controller design that involves real hardware. |
| title | Time-Varying Convex Optimization: A Contraction and Equilibrium Tracking Approach |
| topic | Optimization and Control Systems and Control Signal Processing |
| url | https://arxiv.org/abs/2305.15595 |