面向经典根系上的离散凸分析
Towards discrete convex analysis over classical root systems
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中文总结 AI 辅助
本文将离散凸分析从A型整数格拓展至所有经典根系，构建了对应根系结构的L-凸与M-凸函数理论，证明其局部最优性保证全局最优性，并建立两类函数的离散Fenchel-Legendre共轭一一对应关系。
中文摘要 AI 辅助
离散凸分析（Discrete Convex Analysis, DCA）是连续凸分析的离散类似物，最初被提出作为可高效求解的组合优化问题的统一理论框架。近期，DCA已被证明是涵盖运筹学、经济学及纯数学等多领域的强大工具。受DCA广泛适用性的启发，本文在由经典根系导出的离散结构上建立了离散凸分析的统一理论，拓展了通常对应A型的整数格常规设定。本文采用欧几里得 Coxeter 复形的顶点集作为L-凸性的原始离散域，根格作为M-凸性的对偶离散域；利用相关多面体结构，本文构建了L-凸函数与M-凸函数，以及由根系确定的整性概念，证明这类函数的局部最优性可保证全局最优性；此外，本文还证明整L-凸函数与整M-凸函数可通过离散Fenchel-Legendre共轭一一对应，从而将原始DCA中的共轭关系从A型拓展至所有经典根系。
英文摘要
Discrete Convex Analysis (DCA) is a discrete analog of continuous convex analysis, originally proposed as a unified theoretical framework for efficiently solvable combinatorial optimization problems. Recently, DCA has proven to be a powerful tool across diverse fields, ranging from operations research to economics and pure mathematics. Motivated by the broad applicability of DCA, this paper establishes a unified theory of discrete convex analysis over discrete structures arising from classical root systems, extending the usual setting of the integer lattice, which essentially corresponds to type A. We adopt the vertex set of the Euclidean Coxeter complex as the primal discrete domain for L-convexity, and the root lattice as the dual discrete domain for M-convexity. Using the associated polyhedral structures, we formulate L- and M-convex functions together with notions of integrality determined by the root system. We show that local optimality guarantees global optimality for these functions. Furthermore, we establish that integral L-convex functions and integral M-convex functions correspond one-to-one via the discrete Fenchel--Legendre conjugate, thereby extending the conjugacy in the original DCA from type A to all classical root systems.
