In full generality, minimizing a polynomial function over a closed semi-algebraic set requires complex mathematical equations. This book explains recent developments from singularity theory and semi-algebraic geometry for studying polynomial optimization problems. Classes of generic problems are defined in a simple and elegant manner by using only the two basic (and relatively simple) notions of Newton polyhedron and non-degeneracy conditions associated with a given polynomial optimization problem. These conditions are well known in singularity theory, however, they are rarely considered within the optimization community.
Explanations focus on critical points and tangencies of polynomial optimization, Hölderian error bounds for polynomial systems, Frank–Wolfe-type theorem for polynomial programs and well-posedness in polynomial optimization. It then goes on to look at optimization for the different types of polynomials. Through this text graduate students, PhD students and researchers of mathematics will be provided with the knowledge necessary to use semi-algebraic geometry in optimization.
AuthorHuy-VuiHà, AuthorTiến-SơnPhạm0Semi-Algebraic Geometry, Critical Points and Tangencies, Hölderian Error Bounds, Frank–Wolfe Type Theorem, Polynomial Optimization, Genericity, Compact Semi-Algebraic Sets, Noncompact Semialgebraic Sets, Convex Polynomial OptimizationThe book will complement what is already rather extensively covered in the existing recent books, like the ones of Jean Bernard Lasserre, Monique Laurent, and Murray Marshall