Self-Triggered Time-Varying Convex Optimization

TL;DR

This paper introduces a self-triggered algorithm for solving convex optimization problems with time-varying objectives, reducing continuous computation needs by predicting gradient evolution, and guarantees convergence without Zeno behavior.

Contribution

It proposes a novel self-triggered approach that adaptively schedules computations based on predicted gradient evolution for time-varying convex optimization.

Findings

Guarantees convergence to near-optimal solutions in finite time

Prevents Zeno behavior in the self-triggered scheme

Validated through numerical simulations

Abstract

In this paper, we propose a self-triggered algorithm to solve a class of convex optimization problems with time-varying objective functions. It is known that the trajectory of the optimal solution can be asymptotically tracked by a continuous-time state update law. Unfortunately, implementing this requires continuous evaluation of the gradient and the inverse Hessian of the objective function which is not amenable to digital implementation. Alternatively, we draw inspiration from self-triggered control to propose a strategy that autonomously adapts the times at which it makes computations about the objective function, yielding a piece-wise affine state update law. The algorithm does so by predicting the temporal evolution of the gradient using known upper bounds on higher order derivatives of the objective function. Our proposed method guarantees convergence to arbitrarily small neighborhood of the optimal trajectory in finite time and without incurring Zeno behavior. We illustrate our framework with numerical simulations.