  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.
