Convex and Quadratic Programming
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With the exception of Section 3.2, this book is entirely devoted to a single method of solving nonlinear programming problems, namely the linearization method. In this, it differs from most books on this subject, which usually consider various methods. The various algorithms and approaches described in the literature are not random. Many years of experience of solving nonlinear optimization problems have led specialists to the almost unanimous view that it is impossible to develop a universal algorithm, which could be uniformly and successfully applied to solve all problems. The author fully agrees with this view. Indeed, nonlinear optimization problems are extremely varied. They differ in their nonlinear structure, in the number of variables and constraints and in the amount of storage required. Practical experience shows that there exist classes of problems for which, thanks to a simple implementation and the specific nature of individual problems, a method which is apparently very ineffective from a theoretical point of view may give good results. In this respect, overall, practical computations require a number of different algorithms. The concentration of this book on a single technique reflects a desire to explain its properties and capabilities in detail and to identify its practical characteristics and advantages. The increasing practical use of the linearization method over many years has demonstrated that it is highly effective for solving very broad classes of problems. Thus, here, we shall attempt to identify its most characteristic features which provide for its high effectiveness.
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