Cone-Constrained Procrustes Optimization: A Unified Framework for Geometric Alignment and Conic-Constrained Optimization

The Procrustes problem is a classic geometric alignment model in statistics, computer vision, and shape analysis, with the core objective of minimizing the Frobenius distance between two matrices under rotation, reflection, or orthogonal transformations. On the other hand, general cone programming is a unified framework in convex optimization, describing problems such as linear programming, quadratic programming, and semidefinite programming. Although these two types of problems often overlap in applications, they have long been theoretically independent. This paper proposes the **Cone-Constrained Procrustes Optimization (CCPO)** model, unifying the Procrustes geometric alignment problem and cone-constrained optimization within a single framework. This model achieves unified optimization of geometric transformations and convex cone constraints by introducing cone constraint variables into the alignment objective and constructing a bi-layered optimization structure. This paper further derives the KKT conditions and dual form of the model, and provides convergence analysis and complexity estimation, proving its good optimality and theoretical solvability under convex cone conditions. Finally, this paper discusses the potential applications of the model in computer vision, graph embedding, structural biology, and cross-modal representation learning, providing a new optimization tool for geometric data analysis.