Research

Global Optimization in Model-Based Clustering

We develop several algorithms for the global optimization of the model-based clustering problem, specifically designed for Gaussian clusters. Our motivation is the Expectation-Maximization (EM) algorithm, the most widely known and commonly used algorithm for model-based clustering, is a local optimization method and thus is susceptible to getting trapped in a local solution. The approach is to estimate the parameters of the probability distribution of each cluster, as well as the weights of the clusters, by the maximization of the likelihood function. We apply the Cross-Entropy (CE) method and Model Reference Adaptive Search (MRAS) in our algorithms, relying on the Cholesky decomposition and EM for the updating procedure of the covariance matrices.

Relevant Papers
Heath, J., Fu, M., and Jank, W. New Global Optimization Algorithms for Model-Based Clustering. Working paper. (.pdf)
Heath, J., Fu, M., and Jank, W. Global Convergence of Model Reference Adaptive Search in Gaussian Mixtures. Working paper. (.pdf)

Network Optimization

Collaborators: Damon Gulczynski, Carter Price
Advisor: Bruce Golden

We discuss the Close-Enough Traveling Salesman Problem, a formulation of the TSP in which the salesman's tour needs to pass within a specified radius of each node. We propose several novel heuristics to this problem.

Relevant Paper
Gulczynski, D., Heath, J., and Price, C. Close-Enough Traveling Salesman Problem: A Discussion of Several Heuristics, "Perspectives in Operations Research: Papers in Honor of Saul Gass' 80th Birthday". Springer (2006): 271-283. (.pdf)

Scientific Software

Collaborator: Jason Blevins
Advisor: Willy Hereman

J. Blevins, J. Heath, and W. Hereman, PDESolutionTester.m, 2002.

Description: A Mathematica program for the symbolic verification of exact solutions of nonlinear partial differential equations.

Files: Mathematica Package (.m), Demo Notebook (.nb)

D. Baldwin, Ü. Götkas, W. Hereman, L. Hong, R. Martino, and J.C. Miller. PDESpecialSolutions.m, 2002.

Description: A Mathematica program for the symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for systems of nonlinear partial differential equations.

Files: Directory Listing

Note: For more information and related papers see Dr. Willy Hereman's Home Page.