An ideal market mechanism allocates resources efficiently such that welfare is maximized and sets prices in a way so that the outcome is in a competitive equilibrium and no participant wants to deviate. An important part of the literature discusses Walrasian equilibria and conditions for their existence. We use duality theory to investigate existence of Walrasian equilibria and optimization algorithms to describe auction designs for different market environments in a consistent mathematical framework that allows us to classify the key contributions in the literature and open problems. We focus on auctions with indivisible goods and prove that the relaxed dual winner determination problem is equivalent to the minimization of the Lyapunov function. This allows us to describe central auction designs from the literature in the framework of primal‐dual algorithms. We cover important properties for existence of Walrasian equilibria derived from discrete convex analysis, and provide open research questions.

中文翻译：

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从优化的角度看瓦尔拉斯均衡：文献指南

理想的市场机制可以有效地分配资源，从而使福利最大化并以某种方式设定价格，从而使结果处于竞争均衡状态，并且没有参与者希望偏离。文献的重要部分讨论了瓦尔拉斯均衡及其存在的条件。我们使用对偶理论来研究Walrasian均衡的存在，并使用优化算法在一个一致的数学框架中描述不同市场环境的拍卖设计，这使我们能够对文献中的关键贡献和未解决的问题进行分类。我们专注于不可分割商品的拍卖，并证明宽松的双赢者确定问题等同于Lyapunov函数的最小化。这使我们能够在原始对偶算法的框架下从文献中描述中央拍卖设计。我们涵盖了从离散凸分析得出的Walrasian平衡存在的重要性质，并提供了开放的研究问题。