This second edition of the book includes a thorough theoretical and computational analysis of unconstrained and constrained optimization algorithms. The qualifier modern in the title refers to the unconstrained and constrained optimization algorithms that combine and integrate the latest and the most efficient optimization techniques and advanced computational linear algebra methods. A prime concern of this book is to understand the nature, purposes and limitations of modern nonlinear optimization algorithms. This clear, friendly and rigorous exposition discusses in an axiomatic manner the theory behind the nonlinear optimization algorithms for understanding their properties and their convergence. The presentation of the computational performances of the most known modern nonlinear optimization algorithms is a priority. The book is designed for self-study by professionals or undergraduate or graduate students with a minimal background in mathematics, including linear algebra, calculus, topology and convexity. It is addressed to all those interested in developing and using new advanced techniques for solving large-scale unconstrained or constrained complex optimization problems. Mathematical programming researchers, theoreticians and practitioners in operations research, practitioners in engineering and industry researchers, as well as graduate students in mathematics, Ph.D. and master in mathematical programming will find plenty of recent information and practical approaches for solving real large-scale optimization problems and applications.