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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Rural planning is an important part of territorial spatial planning. The study of rural “Ecological-production-living Space” is the key part in optimizing the land spatial development and improving the spatial planning system, which is significant for the sustainable planning and for high efficiency utilization of rural areas. In this study we take Shuihu village, Hengdong County, China, as a case study. (1) The paper applies the Analytic Hierarchy Process to construct a multi-factor evaluation system of villages “Ecological-production-living Space” for their development and construction, agricultural production conditions and ecological function suitability. Establishing the hierarchy model of rural “Ecological-production-living Space” and affecting weight values to each evaluation factor. (2) Using GIS multi-factor grid, we comprehensively evaluate the village space and identify as well as its dominant function. In the plan, the agricultural production as the leading function and as ecological and living space accounts for 36.25% of the area of the village field. The ecological production accounted for 51.71%, the development and construction accounts for 9.67%. (3) Apply rural complex concept to guide space usage separately from the production space (production and living space) , ecological space (ecological living and ecological production space), living space (living ecology and production space) ecological production and living space, to improve the ecological conservation and energy efficiency of agricultural production, to build ecological agriculture, modern agriculture and liveable idyllic community. Through these above parts, we hope to achieve the sustainable development model of rural area and high efficiency utilization of land resources.