Competitive Online Optimization with Multiple Inventories

Abstract

We study an online inventory trading problem where a user seeks to maximize the aggregate revenue of trading multiple inventories over a time horizon. The trading constraints and concave revenue functions are revealed sequentially in time, and the user needs to make irrevocable decisions. The problem has wide applications in various engineering domains. Existing works employ the primal-dual framework to design online algorithms with sub-optimal, albeit near-optimal, competitive ratios (CR). We exploit the problem structure to develop a new divide-and-conquer approach to solve the online multi-inventory problem by solving multiple calibrated single-inventory ones separately and combining their solutions. The approach achieves the optimal CR of łn θ + 1 if Nłeq łn θ + 1, where N is the number of inventories and θ represents the revenue function uncertainty; it attains a CR of 1/[1-e^-1/(łnθ+1) ] in [łn θ +1, łn θ +2) otherwise. The divide-and-conquer approach reveals novel structural insights for the problem, (partially) closes a gap in existing studies, and generalizes to broader settings. For example, it gives an algorithm with a CR within a constant factor to the lower bound for a generalized one-way trading problem with price elasticity with no previous results. When developing the above results, we also extend a recent CR-Pursuit algorithmic framework and introduce an online allocation problem with allowance augmentation, both of which can be of independent interest.

Online Allocation with Replenishable Budgets: Worst Case and Beyond

“…The assumptions are standard in the literature on online allocation with budget constraints [9,11,40]. Note that we do not require concavity of the utility functions, making our algorithms applicable for a wide range of applications.…”

“…Online allocation with budget constraints in adversarial settings is very challenging and has not been fully resolved yet. Concretely, for online allocation with inventory constraints, competitive online algorithms are designed by pursuing a pseudo-optimal algorithm, but the utility function either takes a single scalar [41] or is separable over multiple dimensions [40]. A recent study [9] considers online allocation with a more general convex utility function and proposes dual mirror descent (DMD) to update the Lagrangian multiplier given stochastic inputs at each round, while the extension to adversarial settings has been considered more recently in [11] and extension to uncertain time horizons is studied in [8].…”

“…-norms (𝑝 ≥ 1). ), and thus get a sufficient condition for the robust constriant(40) as +1 − 𝐵 • 𝑚,𝑡 +1 | + ≥ (𝜆 − 1) 𝑡 ∑︁ 𝑖=1 𝑓 𝑖 (𝑥 • 𝑖 ) − 𝑅. (41)By(39) and the monotonicity of ReLU operation, we have|𝐵 † 𝑚,𝑡 +1 − 𝐵 • 𝑚,𝑡 +1 | + ≤ |𝐵 † 𝑚,𝑡 +1 − 𝛾 B * 𝑚,𝑡 +1 − (1 − 𝛾)𝐵 † 𝑚,𝑡 +1 | + = 𝛾 |𝐵 † 𝑚,𝑡 +1 − B * 𝑚,𝑡 +1 | + .…”

“…First, we propose two novel online algorithms OACP and OACP+ that achieve bounded asymptotic competitive ratios for online allocation with replenishable budgets in adversarial settings (Theorem 3.1 and Theorem 3.2). To our knowledge, the proposed provably-competitive algorithms advance the existing competitive online algorithms to address budget replenishment in adversarial settings for the first time [11,40,41]. Second, we move beyond the worst case and propose a novel learning-augmented algorithm, LA-OACP, that probably improves the average utility compared to OACP or OACP+ (Theorem 4.2), while still offering worst-case utility guarantees for online allocation with budget replenishment for any problem instance (Theorem 4.1).…”

“…(41)By(39) and the monotonicity of ReLU operation, we have|𝐵 † 𝑚,𝑡 +1 − 𝐵 • 𝑚,𝑡 +1 | + ≤ |𝐵 † 𝑚,𝑡 +1 − 𝛾 B * 𝑚,𝑡 +1 − (1 − 𝛾)𝐵 † 𝑚,𝑡 +1 | + = 𝛾 |𝐵 † 𝑚,𝑡 +1 − B * 𝑚,𝑡 +1 | + . (42)Substituting the expressions of 𝑥 • 𝑡 and (42) into the inequality, the sufficient condition for the robust constraint(40) becomes−𝛾𝜆𝐿 𝑡 ∑︁ 𝑖=1 ∥ x * 𝑖 − 𝑥 † 𝑖 ∥ 1 − 𝛾𝜆𝐿 𝑀 ∑︁ 𝑚=1 |𝐵 † 𝑚,𝑡 +1 − B * 𝑚,𝑡 +1 | + ≥ (𝜆 − 1) 𝑡 ∑︁ 𝑖=1 𝑓 𝑖 (𝑥 • 𝑖 ) − 𝑅. (43)By the definition of 𝐵 † 𝑡 and B * 𝑡 , we have + Ê𝑚,𝑡 , 𝐵 max } − 𝑥 † 𝑚,𝑡 − min{ B * 𝑚,𝑡 + Ê𝑚,𝑡 , 𝐵 max } − x *…”