Introductory Lectures on Convex Optimization: A Basic Course: 87 (Applied Optimization)

Nesterov, Yurii

ISBN 10: 1461346916 ISBN 13: 9781461346913

Verlag: Springer, 2013

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IT WAS IN THE MIDDLE OF THE 1980S, WHEN THE SEMINAL PAPER BY KAR� MARKAR OPENED A NEW EPOCH IN NONLINEAR OPTIMIZATION. THE IMPORTANCE OF THIS PAPER, CONTAINING A NEW POLYNOMIAL-TIME ALGORITHM FOR LINEAR OP� TIMIZATION PROBLEMS, 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. AFTERALMOST FIFTEEN YEARS OF INTENSIVE RESEARCH, THE MAIN 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 [12].

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Rese�a del editor:

It was in the middle of the 1980s, when the seminal paper by Kar� markar opened a new epoch in nonlinear optimization. The importance of this paper, containing a new polynomial-time algorithm for linear op� timization problems, 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. Afteralmost fifteen years of intensive research, the main 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 [12].

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