凸对偶性合同用于生产级数学优化
Convex duality contracts for production-grade mathematical optimization
AI总结：
本文提出了一种基于简化弗恩切尔对偶性方案的抽象对偶对，用于统一MathOpt中各类问题的对偶解和不可行性证书，提升优化问题的清晰度和可操作性。
摘要：
在自主生产系统中部署数学优化需要精确的合同来规定优化求解器返回的对象。不幸的是，关于对偶解和不可行性证书（射线）的惯例在求解器和问题类别之间差异很大。本文介绍了MathOpt（一个由谷歌开发和使用的专用语言）所使用的理论框架，用于统一这些概念。我们提出了一种基于简化弗恩切尔对偶性方案的抽象对偶对，允许对所有目前由MathOpt支持的问题类别（包括具有线性和二次目标函数以及线性、锥形、二次和双侧线性约束的问题）进行机械推导的对偶问题和相关合同。我们还展示了这些合同如何提高某些问题类别的互补松弛最优条件的清晰度。
英文摘要：
Deploying mathematical optimization in autonomous production systems requires precise contracts for objects returned by an optimization solver. Unfortunately, conventions on dual solution and infeasibility certificates (rays) vary widely across solvers and classes of problems. This paper presents the theoretical framework used by MathOpt (a domain-specific language developed and used at Google) to unify these notions. We propose an abstract primal-dual pair based on a simplified Fenchel duality scheme that allows for the mechanical derivation of dual problems and associated contracts for all classes of problems currently supported by MathOpt (including those with linear and quadratic objectives plus linear, conic, quadratic, and two-sided linear constraints). We also show how these contracts can improve clarity of complementary-slackness based optimality conditions for certain classes of problems.
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