#### A Lyapunov-Based Methodology for Constrained Optimization with Bandit Feedback

##### Semih Cayci, Yilin Zheng, Atilla Eryılmaz

In a wide variety of applications including online advertising, contractual hiring, and wireless scheduling, the controller is constrained by a stringent budget constraint on the available resources, which are consumed in a random amount by each action, and a stochastic feasibility constraint that may impose important operational limitations on decision-making. In this work, we consider a general model to address such problems, where each action returns a random reward, cost, and penalty from an unknown joint distribution, and the decision-maker aims to maximize the total reward under a budget constraint $B$ on the total cost and a stochastic constraint on the time-average penalty. We propose a novel low-complexity algorithm based on Lyapunov optimization methodology, named ${\tt LyOn}$, and prove that it achieves $O(\sqrt{B\log B})$ regret and $O(\log B/B)$ constraint-violation. The low computational cost and sharp performance bounds of ${\tt LyOn}$ suggest that Lyapunov-based algorithm design methodology can be effective in solving constrained bandit optimization problems.

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