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This paper considers time-average optimization, where a decision vector ischosen every time step within a (possibly non-convex) set, and the goal is tominimize a convex function of the time averages subject to convex constraintson these averages. Such problems have applications in networking, multi-agentsystems, and operations research, where decisions are constrained to a discreteset and the decision average can represent average bit rates or average agentactions. This time-average optimization extends traditional convex formulationsto allow a non-convex decision set. This class of problems can be solved byLyapunov optimization. A simple drift-based algorithm, related to a classicaldual subgradient algorithm, converges to an $\epsilon$-optimal solution within$O(1/\epsilon^2)$ time steps. Further, the algorithm is shown to have atransient phase and a steady state phase which can be exploited to improveconvergence rates to $O(1/\epsilon)$ and $O(1/{\epsilon^{1.5}})$ when vectorsof Lagrange multipliers satisfy locally-polyhedral and locally-smoothassumptions respectively. Practically, this improved convergence suggests thatdecisions should be implemented after the transient period.

Time-Average Optimization with Non-Convex Decision Set and Its Convergence