Jon Lee

This monograph presents a comprehensive treatment of the maximum-entropy sampling problem (MESP), which is a fascinating topic at the intersection of mathematical optimization and data science. The text situates MESP in information theory, as the algorithmic problem of calculating a sub-vector of pre-specificed size from a multivariate Gaussian random vector, so as to maximize Shannon's differential entropy. The text collects and expands on state-of-the-art algorithms for MESP, and addresses its application in the field of environmental monitoring. While MESP is a central optimization problem in the theory of statistical designs (particularly in the area of spatial monitoring), this book largely focuses on the unique challenges of its algorithmic side. From the perspective of mathematical-optimization methodology, MESP is rather unique (a 0/1 nonlinear program having a nonseparable objective function), and the algorithmic techniques employed are highly non-standard. In particular, successful techniques come from several disparate areas within the field of mathematical optimization; for example: convex optimization and duality, semidefinite programming, Lagrangian relaxation, dynamic programming, approximation algorithms, 0/1 optimization (e.g., branch-and-bound), extended formulation, and many aspects of matrix theory. The book is mainly aimed at graduate students and researchers in mathematical optimization and data analytics.

This monograph presents a comprehensive treatment of the maximum-entropy sampling problem (MESP), which is a fascinating topic at the intersection of mathematical optimization and data science. The text situates MESP in information theory, as the algorithmic problem of calculating a sub-vector of pre-specificed size from a multivariate Gaussian random vector, so as to maximize Shannon's differential entropy. The text collects and expands on state-of-the-art algorithms for MESP, and addresses its application in the field of environmental monitoring. While MESP is a central optimization problem in the theory of statistical designs (particularly in the area of spatial monitoring), this book largely focuses on the unique challenges of its algorithmic side. From the perspective of mathematical-optimization methodology, MESP is rather unique (a 0/1 nonlinear program having a nonseparable objective function), and the algorithmic techniques employed are highly non-standard. In particular, successful techniques come from several disparate areas within the field of mathematical optimization; for example: convex optimization and duality, semidefinite programming, Lagrangian relaxation, dynamic programming, approximation algorithms, 0/1 optimization (e.g., branch-and-bound), extended formulation, and many aspects of matrix theory. The book is mainly aimed at graduate students and researchers in mathematical optimization and data analytics.

A First Course in Combinatorial Optimization is a text for a one-semester introductory graduate-level course for students of operations research, mathematics, and computer science. It is a self-contained treatment of the subject, requiring only some mathematical maturity. Topics include: linear and integer programming, polytopes, matroids and matroid optimization, shortest paths, and network flows. Central to the exposition is the polyhedral viewpoint, which is the key principle underlying the successful integer-programming approach to combinatorial-optimization problems. Another key unifying topic is matroids. The author does not dwell on data structures and implementation details, preferring to focus on the key mathematical ideas that lead to useful models and algorithms. Problems and exercises are included throughout as well as references for further study.