Competitive Online Optimization under Inventory Constraints

Abstract

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.

Competitive Algorithms for the Online Multiple Knapsack Problem with Application to Electric Vehicle Charging

“…• We design an algorithm for the general FOMKP problem with rate constraints that has a competitive ratio within an additive factor of one from the optimal competitive ratio. The algorithm also matches or improves upon best-known results in specific cases covered by recent papers, e.g., [25,44,45,48]. • We illustrate the performance of the algorithm in the context of EV charging using a tracebased case study, showing a decrease in the worst case by up to 44% and of the average case by around 20% as compared to fixed threshold policies, the most common approach in prior work on online knapsack problems.…”

“…For example, versions of online 0/1 knapsack [45], online multiple knapsacks, where items can be assigned across multiple knapsacks [48], and online fractional knapsack, where each item can be partially admitted [31]. Similarly, a wide set of variants of one-way trading have emerged, e.g., with [15] or without [44] leftover assets, and concave returns [25]. The disconnected nature of these literatures begs the question: Is it possible for a unified algorithmic approach to be developed or does each variant truly require a carefully crafted approach?…”

Information Aggregation for Constrained Online Control

“…In the remainder of the paper, we sometimes write Regret( ), in replacement of Regret(u) defined in (3), with = (c : + −1 , : + −1 (•|u < )). It is worth mentioning that, to the best of our knowledge, there is no existing bound on the dynamic regret defined in (3) in the current setting, nor are there existing results for a similar worst-case metric called competitive ratio [11,[34][35][36]43].…”

“…Example 1 (Inventory constraints). Consider the following set of inventory constraints, which is a common form of constraints in energy storage and demand response problems [30,35,45]. The summation of the squares of the ℓ 2 norms of the actions is bounded from above by > 0, representing the limited resources of the system: =1 || || 2 2 ≤ where ∈ R .…”

“…Designing controllers for the general constrained dynamics in ( 1) is challenging and, as a result, traditional online optimization models often adopt simplified versions of (1), such as unconstrained optimization [11,22,38] or time-invariant constraints [26,27] that are known a priori. More specifically, previous studies in online control and online optimization mostly focus on specific forms of constraints or costs depending on different applications, such as switching costs [17,33,35,43], ramping constraints [4,43], polytopic constraints [16], time-varying memoryless cumulative constraints [9,46,[48][49][50], convex loss functions with memory [1][2][3]42] or inventory constraints [30,35]. Within this literature, the goal is to derive policies with either small regret [9,15,33,35,49,51] or competitive ratio [4,17,34,42,43].…”