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

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 but 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 ($\texttt{OIPM-TEC}$). We derive simultaneous sublinear dynamic regret and constraint violation bounds for $\texttt{OIPM-TEC}$ under standard assumptions. For applications where a given tolerance around optima is accepted, we propose a new OCO performance metric -- the epsilon-regret -- and a more computationally efficient algorithm, the epsilon $\texttt{OIPM-TEC}$, 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.