Time-Varying Convex Optimization: A Contraction and Equilibrium Tracking Approach
arxiv(2023)
摘要
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 with proportionality constant upper bounded by Lipschitz constant in
which the parameter appears divided by the contraction rate of the dynamics
squared. We additionally establish that any parameter-dependent contracting
dynamics can be augmented with a feedforward prediction term to ensure that the
tracking error converges to zero exponentially quickly. To apply these results
to time-varying convex optimization problems, we establish the strong
infinitesimal contractivity of dynamics solving three canonical problems,
namely monotone inclusions, linear equality-constrained problems, and composite
minimization problems. For each of these problems, we prove the sharpest-known
rates of contraction and provide explicit tracking error bounds between
solution trajectories and minimizing trajectories. We validate our theoretical
results on three numerical examples including an application to control-barrier
function based controller design.
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