Prediction-Correction Interior-Point Method for Time-Varying Convex Optimization

TL;DR

This paper introduces a novel interior-point method using a continuous-time dynamical system to efficiently track solutions of convex optimization problems with time-varying objectives and constraints, ensuring exponential convergence.

Contribution

It proposes a new prediction-correction interior-point approach with a continuous-time dynamical system for time-varying convex optimization, achieving global exponential convergence.

Findings

Method successfully tracks time-varying solutions with vanishing error.

Convergence to optimal solution is exponential.

Applicable to practical problems like sparse least squares and robot navigation.

Abstract

In this paper, we develop an interior-point method for solving a class of convex optimization problems with time-varying objective and constraint functions. Using log-barrier penalty functions, we propose a continuous-time dynamical system for tracking the (time-varying) optimal solution with an asymptotically vanishing error. This dynamical system is composed of two terms: (i) a correction term consisting of a continuous-time version of Newton's method, and (ii) a prediction term able to track the drift of the optimal solution by taking into account the time-varying nature of the objective and constraint functions. Using appropriately chosen time-varying slack and barrier parameters, we ensure that the solution to this dynamical system globally asymptotically converges to the optimal solution at an exponential rate. We illustrate the applicability of the proposed method in two practical applications: a sparsity promoting least squares problem and a collision-free robot navigation problem.