Prediction-Correction Algorithms for Time-Varying Constrained Optimization

TL;DR

This paper introduces online prediction-correction algorithms for tracking solutions of time-varying constrained optimization problems, improving convergence speed and applicability to constrained scenarios without requiring Hessian inverses.

Contribution

It extends existing prediction-correction methods to constrained problems and develops first-order prediction steps that do not rely on Hessian inverses.

Findings

Proposed algorithms effectively track optimal solutions in simulations.

Methods improve convergence speed over existing approaches.

Application demonstrated in real-time energy resource control.

Abstract

This paper develops online algorithms to track solutions of time-varying constrained optimization problems. Particularly, resembling workhorse Kalman filtering-based approaches for dynamical systems, the proposed methods involve prediction-correction steps to provably track the trajectory of the optimal solutions of time-varying convex problems. The merits of existing prediction-correction methods have been shown for unconstrained problems and for setups where computing the inverse of the Hessian of the cost function is computationally affordable. This paper addresses the limitations of existing methods by tackling constrained problems and by designing first-order prediction steps that rely on the Hessian of the cost function (and do not require the computation of its inverse). In addition, the proposed methods are shown to improve the convergence speed of existing prediction-correction methods when applied to unconstrained problems. Numerical simulations corroborate the analytical results and showcase performance and benefits of the proposed algorithms. A realistic application of the proposed method to real-time control of energy resources is presented.