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We study two canonical online optimization problems under capacity/budget constraints: the fractional one-way trading problem (OTP) and the integral online knapsack problem (OKP) under an infinitesimal assumption. Under the competitive analysis framework, it is well-known that both problems have the same optimal competitive ratio. However, these two problems are investigated by distinct approaches under separate contexts in the literature. There is a gap in understanding the connection between these two problems and the nature of their online algorithm design. This paper provides a unified framework for the online algorithm design, analysis and optimality proof for both problems. We find that the infinitesimal assumption of the OKP is the key that connects the OTP in the analysis of online algorithms and the construction of worst-case instances. With this unified understanding, our framework shows its potential for analyzing other extensions of OKP and OTP in a more systematic manner.

Optimal Online Algorithms for One-Way Trading and Online Knapsack Problems: A Unified Competitive Analysis