Randomized rounding is a technique that was originally used to approximate hard offline discrete optimization problems from a mathematical programming relaxation. Since then it has also been used to approximately solve sequential stochastic optimization problems, overcoming the curse of dimensionality. To elaborate, one first writes a tractable linear programming relaxation that prescribes probabilities with which actions should be taken. Rounding then designs a (randomized) online policy that approximately preserves all of these probabilities, with the challenge being that the online policy faces hard constraints, whereas the prescribed probabilities only have to satisfy these constraints in expectation. Moreover, unlike classical randomized rounding for offline problems, the online policy's actions unfold sequentially over time, interspersed by uncontrollable stochastic realizations that affect the feasibility of future actions. This tutorial provides an introduction for using randomized rounding to design online policies, through four self-contained examples representing fundamental problems in the area: online contention resolution, stochastic probing, stochastic knapsack, and stochastic matching.

翻译：随机化舍入是一种最初用于从数学规划松弛中逼近困难离线离散优化问题的技术。此后，该方法也被用于近似求解序列随机优化问题，以克服维度灾难问题。具体而言，首先构建一个易处理的线性规划松弛，该松弛规定了应采取行动的概率分布。随后通过舍入设计一个（随机化的）在线策略，以近似保持所有这些概率，其挑战在于在线策略面临硬性约束，而规定的概率仅需在期望意义上满足这些约束。此外，与离线问题的经典随机化舍入不同，在线策略的行动随时间顺序展开，其间穿插着不可控的随机实现，这些实现会影响后续行动的可行性。本教程通过该领域的四个基础问题——在线竞争消解、随机探测、随机背包与随机匹配——的自包含示例，系统介绍如何利用随机化舍入设计在线策略。