Online Convex Optimization with Binary Constraints
This addresses online optimization with binary constraints for applications like demand response, presenting an incremental algorithmic improvement.
The paper tackles online convex optimization with binary decision variables by proposing binary online gradient descent (bOGD), achieving sublinear regret bounds that depend on the cumulative variation of relaxed optima. It demonstrates application to demand response with thermostatically controlled loads, showing randomization does not significantly degrade performance in simulations.
We consider online optimization with binary decision variables and convex loss functions. We design a new algorithm, binary online gradient descent (bOGD) and bound its expected dynamic regret. We provide a regret bound that holds for any time horizon and a specialized bound for finite time horizons. First, we present the regret as the sum of the relaxed, continuous round optimum tracking error and the rounding error of our update in which the former asymptomatically decreases with time under certain conditions. Then, we derive a finite-time bound that is sublinear in time and linear in the cumulative variation of the relaxed, continuous round optima. We apply bOGD to demand response with thermostatically controlled loads, in which binary constraints model discrete on/off settings. We also model uncertainty and varying load availability, which depend on temperature deadbands, lockout of cooling units and manual overrides. We test the performance of bOGD in several simulations based on demand response. The simulations corroborate that the use of randomization in bOGD does not significantly degrade performance while making the problem more tractable.