The goal of this monograph is to present in a rigorous mathematical frame the main aspects of convergence of the optimization algorithms. The convergence of the Newton method for unconstrained optimization, the ellipsoid and the general interior point methods for linear programming, as well as the barrier and cutting plane methods for constrained optimization are considered.

    The monograph is structured în 8 chapters. The first one has an introductory character giving some insights into standard optimization problems as well as into the standard problems of numerical linear algebra. The next three chapters present in a very general manner the fundamental concepts of global and asimptotic convergence. The main part of the monograph is represented by chapters 5, 6 and 7. Chapter 5 considers the convergence analysis of unconstrained optimization algorithms. The descent gradient method and the Newton method with backtracking are looked upon in details. A special section is dedicated to the analysis of the Newton method for self-concordant functions. It is shown that the number of iterations corresponding to the damped Newton algorithm with backtracking linear searching depends on the initial suboptimality, as well as on a number of (unknown) parameters associated with the function and its Hessian. Chapter 6 is for convergence analysis of the simplex method, the ellipsoid method and the path-following interior point methods for linear programming. In chapter 7 the convergence analysis of the barrier method and of the cutting plane method for constrained optimization are presented. For the self-concordant, strong convex barrier functions it is shown that the number of Newton iterations is a function of the number of inequalities, of the parameter of linear searching with backtracking, as well as of some other known constants. The last chapter includes some considerations on theory versus empiricism in the analysis of optimization algorithms. The veritable algorithmist always swings between pure theoretical developments and experimental ones, being aware that neither of them is superior to each other, they only complete each other and only their joined results can eventually lead to the understanding of the efficiency and limits of algorithms.

    The book has been written to serve a diverse audience of scientists, engineers and decision makers, researchers, computer scientists or professionals working in academia, business, industry and government facing design optimization models and algorithms.
 

                                                                                                                                                   Neculai Andrei