# Time-Varying Convex Optimization: A Contraction and Equilibrium Tracking Approach

arXiv: https://arxiv.org/abs/2305.15595 · HTML: https://mlsanity.com/paper/2305.15595/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 May 2023.

## Built on by (most cited first, 6 of 6)

- [An Energy-Based Mechanism for Compositional Behavior](https://mlsanity.com/paper/2512.04745/) (2025)
- [Contractivity and linear convergence in bilinear saddle-point problems: An operator-theoretic approach](https://mlsanity.com/paper/2410.14592/) (2024)
- [Small-Gain Nash: Certified Contraction to Nash Equilibria in Differentiable Games](https://mlsanity.com/paper/2512.06791/) (2025)
- [A Nonlinear Separation Principle via Contraction Theory: Applications to Neural Networks, Control, and Learning](https://mlsanity.com/paper/2604.15238/) (2026)
- [Predictive Coding with Bayesian Priors via Proximal Gradients](https://mlsanity.com/paper/2606.08374/) (2026)
- [Constrained Flow Matching via Lagrangian Dual Flows](https://mlsanity.com/paper/2607.04513/) (2026)

## Abstract (arXiv)

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.