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This paper considers time-average optimization, where a decision vector is chosen every time step within a (possibly nonconvex) set, and the goal is to minimize a convex function of the time averages subject to convex constraints on these averages. Such problems have applications in networking, multiagent systems, and operation research, where decisions are constrained to a discrete set and the decision average can represent average bit rates or average agent actions. This time-average optimization extends traditional convex formulations to allow a nonconvex decision set. This class of problems can be solved by Lyapunov optimization. A simple drift-based algorithm, related to a classical dual subgradient algorithm, converges to an &#x03B5;-optimal solution within O(1/&#x03F5;<sup>2</sup>) time steps. Furthermore, the algorithm is shown to have a transient phase and a steady-state phase, which can be exploited to improve convergence rates to O(1/&#x03F5;) and O(1/&#x03F5;<sup>1.5</sup> ) when vectors of Lagrange multipliers satisfy locally polyhedral and locally smooth assumptions, respectively. Practically, this improved convergence suggests that decisions should be implemented after the transient period.

Time-Average Optimization With Nonconvex Decision Set and Its Convergence