# Online Convex Optimization with Binary Constraints

arXiv: https://arxiv.org/abs/2005.02274 · HTML: https://mlsanity.com/paper/2005.02274/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 May 2020.

## Built on by (most cited first, 4 of 4)

- [Dynamic Regret of Online Mirror Descent for Relatively Smooth Convex Cost Functions](https://mlsanity.com/paper/2202.12843/) (2022)
- [Multi-Agent Reinforcement Learning for Fast-Timescale Demand Response of Residential Loads](https://mlsanity.com/paper/2301.02593/) (2023)
- [CHOMET: Conditional Handovers via Meta-Learning](https://mlsanity.com/paper/2507.07581/) (2025)
- [Online Dynamic Submodular Optimization](https://mlsanity.com/paper/2306.10835/) (2023)

## Abstract (arXiv)

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.