Studying Constrained Clustering Problems Using Homotopy Maps

Probability-one homotopy maps for tracking constrained clustering solutions

Modern machine learning problems typically have multiple criteria, but there is currently no systematic mathematical theory to guide the design of formulations and exploration of alternatives. Homotopy methods are a promising approach to characterize solution spaces by smoothly tracking solutions from one formulation (typically an “easy” problem) to another (typically a “hard” problem). New res...

Probability-one Homotopy Maps for Constrained Clustering Problems

Many algorithms for constrained clustering have been developed in the literature that aim to balance vector quantization requirements of cluster prototypes against the discrete satisfaction requirements of constraint (must-link or cannot-link) sets. A significant amount of research has been devoted to designing new algorithms for constrained clustering and understanding when constraints help cl...

Solution of Constrained Clustering Problems through Homotopy Tracking

Theory of Globally Convergent Probability-One Homotopies for Nonlinear Programming

For many years globally convergent probability-one homotopy methods have been remarkably successful on difficult realistic engineering optimization problems, most of which were attacked by homotopy methods because other optimization algorithms failed or were ineffective. Convergence theory has been derived for a few particular problems, and considerable fixed point theory exists, but generally ...

The Pontrjagin-Hopf invariants for Sobolev maps

Variational problems with topological constraints can exhibit many interesting mathematical properties. There are many open mathematical questions in this area. The diverse properties these problems possess give them the potential for different applications. We find variational problems with topological constraints in high energy physics, hydrodynamics, and material science, [9, 3, 48]. Usually...