Online Interior-point Methods for Time-varying Equality-constrained Optimization

TL;DR

This paper introduces the first projection-free online interior-point method capable of handling time-varying linear constraints and generalized inequalities, with proven sublinear regret and constraint violation bounds, applied to online power flow problems.

Contribution

The paper presents novel online interior-point algorithms that manage time-varying constraints without projections, extending OCO applicability to dynamic, safety-critical systems.

Findings

Achieved sublinear dynamic regret and constraint violation bounds.

Demonstrated effectiveness on online optimal power flow problems.

Compared favorably with existing OCO algorithms.

Abstract

An important challenge in the online convex optimization (OCO) setting is to incorporate generalized inequalities and time-varying constraints. The inclusion of constraints in OCO widens the applicability of such algorithms to dynamic and safety-critical settings such as the online optimal power flow (OPF) problem. In this work, we propose the first projection-free OCO algorithm admitting time-varying linear constraints and convex generalized inequalities: the online interior-point method for time-varying equality constraints (OIPM-TEC). We derive simultaneous sublinear dynamic regret and constraint violation bounds for OIPM-TEC under standard assumptions. For applications where a given tolerance around optima is accepted, we employ an alternative OCO performance metric -- the epsilon-regret -- and a more computationally efficient algorithm, the epsilon-OIPM-TEC, that possesses sublinear bounds under this metric. Finally, we showcase the performance of these two algorithms on an online OPF problem and compare them to another OCO algorithm from the literature.