We consider convex-concave saddle point problems, and more generally convex optimization problems we refer to as $\textit{saddle problems}$, which include the partial supremum or infimum of convex-concave saddle functions. Saddle problems arise in a wide range of applications, including game theory, machine learning, and finance. It is well known that a saddle problem can be reduced to a single convex optimization problem by dualizing either the convex (min) or concave (max) objectives, reducing a min-max problem into a min-min (or max-max) problem. Carrying out this conversion by hand can be tedious and error prone. In this paper we introduce $\textit{disciplined saddle programming}$ (DSP), a domain specific language (DSL) for specifying saddle problems, for which the dualizing trick can be automated. The language and methods are based on recent work by Juditsky and Nemirovski arXiv:2102.01002 [math.OC], who developed the idea of conic-representable saddle point programs, and showed how to carry out the required dualization automatically using conic duality. Juditsky and Nemirovski's conic representation of saddle problems extends Nesterov and Nemirovski's earlier development of conic representable convex problems; DSP can be thought of as extending disciplined convex programming (DCP) to saddle problems. Just as DCP makes it easy for users to formulate and solve complex convex problems, DSP allows users to easily formulate and solve saddle problems. Our method is implemented in an open-source package, also called DSP.
翻译:我们考虑的是Convex- concave马鞍问题,以及更一般的convex优化问题,我们称之为$$(textit{saddle problems}$),其中包括部分超额或最小的 convex-concave 马鞍功能。在一系列广泛的应用中,包括游戏理论、机器学习和金融,出现叠装问题。众所周知,一个马鞍问题可以降低到一个单一的 convex优化问题,方法是将 convex (min) 或容易(max) 的目标双倍化,将一个小麦问题降低到一个最小(或最高马达)问题。用手进行这种转换可能会是乏味和易出错的。在本文中,我们引入了 $\ textitleit{ rolegal sad playproduction} (DSP), 一种域特定语言(DSDL) 来说明马鞍问题,可以实现双重的把戏法自动化。语言和方法基于 Judistosky 和 Nemirovski an (math.