This paper studies online optimization under inventory (budget) constraints. While online optimization is a well-studied topic, versions with inventory constraints have proven difficult. We consider a formulation of inventory-constrained optimization that is a generalization of the classic one-way trading problem and has a wide range of applications. We present a new algorithmic framework, CR-Pursuit, and prove that it achieves the optimal competitive ratio among all deterministic algorithms (up to a problem-dependent constant factor) for inventory-constrained online optimization. Our algorithm and its analysis not only simplify and unify the state-ofthe-art results for the standard one-way trading problem, but they also establish novel bounds for generalizations including concave revenue functions. For example, for one-way trading with price elasticity, CR-Pursuit achieves a competitive ratio within a small additive constant (i.e., 1/3) to the lower bound of ln θ + 1, where θ is the ratio between the maximum and minimum base prices. CCS CONCEPTS • Theory of computation → Online algorithms; Design and analysis of algorithms; • Applied computing → Decision analysis.

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Ride-sharing is a modern urban-mobility paradigm with tremendous potential in reducing congestion and pollution. Demand-aware design is a promising avenue for addressing a critical challenge in ride-sharing systems, namely joint optimization of request-vehicle assignment and routing for a fleet of vehicles. In this paper, we develop a probabilistic demand-aware framework to tackle the challenge. We focus on maximizing the expected number of passenger pickups, given the probability distributions of future demands. The key idea of our approach is to assign requests to vehicles in a probabilistic manner. It differentiates our work from existing ones and allows us to explore a richer design space to tackle the requestvehicle assignment puzzle with a performance guarantee but still keeping the final solution practically implementable. The optimization problem is non-convex, combinatorial, and NP-hard in nature. As a key contribution, we explore the problem structure and propose an elegant approximation of the objective function to develop a dual-subgradient heuristic. We characterize a condition under which the heuristic generates a (1 − 1/e) approximation solution. Our solution is simple and scalable, amendable for practical implementation. Results of numerical experiments based on real-world traces in Manhattan show that, as compared to a conventional demand-oblivious scheme, our demand-aware solution improves the passenger pickups by up to 46%. The results also show that joint optimization at the fleet level leads to 19% more pickups than that by separate optimizations at individual vehicles.