Preface

Simplicial global optimization focuses on deterministic covering methods for global optimization partitioning the feasible region by simplices. Although rectangular partitioning is used most often in global optimization, simplicial covering has advantages shown in this book. The purpose of the book is to present global optimization methods based on simplicial partitioning in one volume. The book describes features of simplicial partitioning and demonstrates its advantages in global optimization. A simplex is a polyhedron in a multidimensional space, which has the minimal number of vertices. Therefore simplicial partitions are preferable in global optimization when the values of the objective function at all vertices of partitions are used to evaluate subregions. The feasible region defined by linear constraints may be covered by simplices and therefore simplicial optimization algorithms may cope with linear constraints in a delicate way by initial covering. This makes simplicial partitions very attractive for optimization problems with linear constraints. There are optimization problems where the objective functions have symmetries which may be taken into account for reducing the search space significantly by setting linear inequality constraints. The resulted search region may be covered by simplices. Applications benefiting from simplicial partitioning are examined in the book: nonlinear least squares regression, center-based clustering of data having one feature, and pile placement in grillage-type foundations. In the examples shown, the search region reduced taking into account symmetries of the objective functions is a simplex thus simplicial global optimization algorithms may use it as a starting partition. The book provides exhaustive experimental investigation and shows the impact of various bounds, types of subdivision, and strategies of candidate selection on the performance of global optimization algorithms. Researchers and engineers will benefit from simplicial partitioning algorithms presented in the book: Lipschitz branch-and-bound, Lipschitz optimization without the Lipschitz constant. We hope