Disciplined Geodesically Convex Programming

• URL: http://arxiv.org/abs/2407.05261v1
• Date: Sun, 7 Jul 2024 05:13:51 GMT
• Title: Disciplined Geodesically Convex Programming
• Authors: Andrew Cheng, Vaibhav Dixit, Melanie Weber,
• Abstract summary: Testing convexity structure in nonlinear programs relies on verifying the convexity objectives and constraints. citetgrantdisciplined introduced a framework, Disciplined Convex Programming (DCP), which can verify basic convexatoms. Our paper is accompanied by a Julia package, which provides functionality for testing DGCP-compliant expressions.
• Score: 0.9899763598214121
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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