Interior Point Method for Dynamic Constrained Optimization in Continuous Time

TL;DR

This paper introduces an interior point method for real-time tracking of solutions in convex optimization problems with continuously changing objectives and constraints, ensuring asymptotic accuracy in dynamic environments.

Contribution

It develops a novel interior point algorithm that incorporates time-varying slack and prediction-correction steps for dynamic constrained optimization.

Findings

Algorithm guarantees asymptotic tracking accuracy.

Method effectively handles multi-agent target tracking scenarios.

Provides theoretical convergence under standard assumptions.

Abstract

This paper considers a class of convex optimization problems where both, the objective function and the constraints, have a continuously varying dependence on time. Our goal is to develop an algorithm to track the optimal solution as it continuously changes over time inside or on the boundary of the dynamic feasible set. We develop an interior point method that asymptotically succeeds in tracking this optimal point in nonstationary settings. The method utilizes a time varying constraint slack and a prediction-correction structure that relies on time derivatives of functions and constraints and Newton steps in the spatial domain. Error free tracking is guaranteed under customary assumptions on the optimization problems and time differentiability of objective and constraints. The effectiveness of the method is illustrated in a problem that involves multiple agents tracking multiple targets.