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

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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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