Convex programming plays a fundamental role in machine learning, data science, and engineering. Testing convexity structure in nonlinear programs relies on verifying the convexity of objectives and constraints. \citet{grant2006disciplined} introduced a framework, Disciplined Convex Programming (DCP), for automating this verification task for a wide range of convex functions that can be decomposed into basic convex functions (atoms) using convexity-preserving compositions and transformations (rules). However, the restriction to Euclidean convexity concepts can limit the applicability of the framework. For instance, many notable instances of statistical estimators and matrix-valued (sub)routines in machine learning applications are Euclidean non-convex, but exhibit geodesic convexity through a more general Riemannian lens. In this work, we extend disciplined programming to this setting by introducing Disciplined Geodesically Convex Programming (DGCP). We determine convexity-preserving compositions and transformations for geodesically convex functions on general Cartan-Hadamard manifolds, as well as for the special case of symmetric positive definite matrices, a common setting in matrix-valued optimization. For the latter, we also define a basic set of atoms. Our paper is accompanied by a Julia package SymbolicAnalysis.jl, which provides functionality for testing and certifying DGCP-compliant expressions. Our library interfaces with manifold optimization software, which allows for directly solving verified geodesically convex programs.

翻译：凸规划在机器学习、数据科学与工程领域发挥着基础性作用。检验非线性规划中的凸性结构依赖于验证目标函数与约束条件的凸性。\citet{grant2006disciplined} 提出了基于规则的凸规划框架，用于对可通过凸性保持的组合与变换规则分解为基本凸函数（原子函数）的广泛凸函数类别实现自动化验证。然而，局限于欧几里得凸性概念会限制该框架的适用性。例如，在机器学习应用中，许多重要的统计估计量与矩阵值（子）程序在欧几里得意义下是非凸的，但通过更一般的黎曼几何视角却展现出测地凸性。在本工作中，我们将基于规则的规划框架扩展至该场景，提出了基于规则的测地凸规划。我们确定了在一般Cartan-Hadamard流形上测地凸函数的凸性保持组合与变换规则，并特别针对对称正定矩阵这一矩阵值优化中的常见情形进行了研究。对于后者，我们还定义了一组基本原子函数集。本文附有Julia软件包SymbolicAnalysis.jl，该工具提供检测与认证符合DGCP规范的表达式的功能。我们的库可与流形优化软件对接，从而能够直接求解已验证的测地凸规划问题。