Probability-one Homotopy Maps for Constrained Clustering Problems

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...

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 ...

Studying Constrained Clustering Problems Using Homotopy Maps

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 must-not-link) sets. Significant research has been devoted to designing new algorithms for constrained clustering and understanding when constraints help clustering. ...

Probability-one homotopy maps for mixed complementarity problems

Probability-one homotopy algorithms have strong convergence characteristics under mild assumptions. Such algorithms for mixed complementarity problems (MCPs) have potentially wide impact because MCPs are pervasive in science and engineering. A probability-one homotopy algorithm for MCPs was developed earlier by Billups and Watson based on the default homotopy mapping. This algorithm had guarant...

Solution of Constrained Clustering Problems through Homotopy Tracking