INTRODUCTORY LECTURES ON CONVEX OPTIMIZATION NESTEROV PDF

A Donald w. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photo-copying, microfilming, recording, or otherwise, without the prior written permission of the publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work PermissionsforbookspublishedintheUsa:permissions wkap. The importance of this paper, containing a new polynomial-time algorithm for linear op- timization problerris, was not only in its complexity bound. At that time, the most surprising feature of this algorithm was that the theoretical pre diction of its high efficiency was supported by excellent computational results. This unusual fact dramatically changed the style and direc- tions of the research in nonlinear optimization.

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A Donald w. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photo-copying, microfilming, recording, or otherwise, without the prior written permission of the publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work PermissionsforbookspublishedintheUsa:permissions wkap.

The importance of this paper, containing a new polynomial-time algorithm for linear op- timization problerris, was not only in its complexity bound. At that time, the most surprising feature of this algorithm was that the theoretical pre diction of its high efficiency was supported by excellent computational results.

This unusual fact dramatically changed the style and direc- tions of the research in nonlinear optimization. Thereafter it became more and more common that the new methods were provided with a complexity analysis, which was considered a better justification of their efficiency than computational experiments.

In a new rapidly develop ing field, which got the name "polynomial-time interior-point methods such a justification was obligatory After almost fifteen years of intensive research, the Inain results of this development started to appear in monographs [12, 14, 16, 17, 18, 19 Approximately at that time the author was asked to prepare a new course on nonlinear optimization for graduate students.

The idea was to create a course which would reflect the new developments in the field. Actually this was a major challenge. At the time only the theory of interior-point methods for linear optimization was polished enough to be explained to students. The general theory of self-concordant functions had appeared in print only once in the form of research monograph [ Moreover it was clear that the new theory of interior-point methods represented only a part of a general theory of convex optimization, a rather involved field with the complexity bounds, optimal methods, etc.

The majority of the latter results were published in different journals in Russian. The book you see now is a result of an attempt to present serious things in an elementary form. In view of a severe volume limitation, we had to be very pragmatic.

Any concept or fact included in the book is absolutely nec essary for the analysis of at least one optimization scheme. Surprisingly enough, none of the material presented requires any facts from duality theory Thus, this topic is completely omitted. This does not mean, of course that the author neglects this fundamental concept.

However, we hope that for the first treatment of the subject such a compromise is acceptable. The main goal of this course is the development of a correct under- standing of the complexity of different optimization problems. This goal was not chosen by chance. Every year I meet Ph. And very often they seem to have come too late. In my experience, if an optimization model is created without taking into account the abilities of numerical schemes the chances that it will be possible to find an acceptable numerical solution are close to zero.

In any field of human activity, if we create something, we know in advance why we are doing so and what we are going to do with the result. And only in numerical modelling is the situation still different This course was given during several years at Universite Catholique de Louvain Louvain-la-Neuve, Belgium. The course is self-contained. It consists of four chapters Nonlinear optimization, Smooth convex opti- mization,Nonsmooth convex optimization and Structural optimization Interior-point methods The chapters are essentially independent and can be used as parts of more general courses on convex analysis or op- timization.

In our experience each chapter can be covered in three two- hour lectures. We assume a reader to have a standard undergraduate background in analysis and linear algebra.

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Introductory Lectures on Convex Optimization: A Basic Course

The final prices may differ from the prices shown due to specifics of VAT rules About this Textbook This book provides a comprehensive, modern introduction to convex optimization, a field that is becoming increasingly important in applied mathematics, economics and finance, engineering, and computer science, notably in data science and machine learning. Written by a leading expert in the field, this book includes recent advances in the algorithmic theory of convex optimization, naturally complementing the existing literature. It contains a unified and rigorous presentation of the acceleration techniques for minimization schemes of first- and second-order. It provides readers with a full treatment of the smoothing technique, which has tremendously extended the abilities of gradient-type methods. Several powerful approaches in structural optimization, including optimization in relative scale and polynomial-time interior-point methods, are also discussed in detail. Researchers in theoretical optimization as well as professionals working on optimization problems will find this book very useful. It presents many successful examples of how to develop very fast specialized minimization algorithms.

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