Linear-programming (LP)-based primal-dual methods are fundamental for designing and analyzing algorithms in adversarial (prior-free) online resource allocation. This chapter provides a tutorial on two modern primal-dual frameworks, emphasizing recent developments and contemporary models in operations research. Part~I develops an LP-based convex-programming framework where solving a regularized convex program at each arrival captures the tradeoff between greediness and hedging, yielding a dual certificate via Karush-Kuhn-Tucker (KKT) conditions. Because standard LP relaxations can be weak or intractable for stochastic outcomes, Part~II introduces a complementary LP-free framework that provides a universal certificate system for evaluating competitive ratios under such uncertainty. Covering a wide array of models -- including online vertex-weighted bipartite matching, edge-weighted online matching with free disposal, online matching with stochastic rewards, reusable resources, two-sided assortment optimization, configuration allocation (whole-page optimization), AdWords, and costly cancellations -- the tutorial equips readers with versatile proof templates to analyze existing algorithms and develop new solutions for emerging applications.

翻译：基于线性规划的原始-对偶方法是设计和分析对抗性（无先验）在线资源分配算法的核心工具。本章提供了一个关于两种现代原始-对偶框架的教程，重点介绍运筹学领域的最新进展与当代模型。第一部分构建了一个基于线性规划的凸规划框架，通过在每个到达时刻求解正则化凸规划，捕捉贪婪性与对冲性之间的权衡，并经由Karush-Kuhn-Tucker条件生成对偶证书。由于标准线性规划松弛在随机结果场景下可能显得薄弱或难以处理，第二部分引入了一种互补的无线性规划框架，为在此类不确定性下评估竞争比提供通用证书系统。本教程涵盖广泛模型——包括在线顶点加权二分图匹配、带自由弃权的边加权在线匹配、带随机奖励的在线匹配、可重用资源、双面品类优化、配置分配（整页优化）、广告词及代价高昂的取消——使读者能够掌握通用的证明范式，用于分析现有算法并为新兴应用开发新方案。