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We study the online decision making problem (ODMP) as a naturalgeneralization of online linear programming. In ODMP, a single decision makerundertakes a sequence of decisions over $T$ time steps. At each time step, thedecision maker makes a locally feasible decision based on information availableup to that point. The objective is to maximize the accumulated reward whilesatisfying some convex global constraints called goal constraints. The decisionmade at each step results in an $m$-dimensional vector that represents thecontribution of this local decision to the goal constraints. In the onlinesetting, these goal constraints are soft constraints that can be violatedmoderately. To handle potential nonconvexity and nonlinearity in ODMP, wepropose a Fenchel dual-based online algorithm. At each time step, the algorithmrequires solving a potentially nonconvex optimization problem over the localfeasible set and a convex optimization problem over the goal set. Under certainstochastic input models, we show that the algorithm achieves $O(\sqrt{mT})$goal constraint violation deterministically, and $\tilde{O}(\sqrt{mT})$ regretin expected reward. Numerical experiments on an online knapsack problem and anassortment optimization problem are conducted to demonstrate the potential ofour proposed online algorithm.

Online Decision Making with Nonconvex Local and Convex Global Constraints