The modern era of interior-point methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadratic programming, semidefinite programming, and nonconvex and nonlinear problems, have reached varying levels of maturity. The first part of the talk will review the most important results in interior-point methods obtained over the past two decades, emphasizing the distinction between computational complexity and superlinear convergence. While the work on computational complexity has shown that interior-point methods can solve in polynomial time some important mathematical programming problems, superlinear convergence results explain why the practical performance of interior-point methods is better than predicted by the computational complexity results. The second part of the talk will concentrate on some recent results obtained by the author and his collaborators.
Florian Potra earned a Ph.D. in Mathematics from the University of Bucharest, Romania. After an Andrew Mellon Postdoctoral Fellowship at the University of Pittsburgh, he joined the faculty of the University of Iowa in 1984, first as an Associate Professor of Mathematics, and then as a Professor of Mathematics and Computer Science. Between 1997-1998, he served as a Program Director in Applied and Computational Mathematics at the National Science Foundation. Since 1998 he has been a Professor of Mathematics and Statistics at the University of Maryland Baltimore County. He is also a Faculty Appointee in the Applied and Computational Mathematics Division of The National Institute of Standards and Technology. Dr. Potra was selected as the Royden B. Davis Chair for Interdisciplinary Studies at Georgetown University for the academic year 2008-2009. Florian Potra was the advisor of 17 Ph.D. students in Applied and Computational Mathematics that are presently employed by academia, industry, and national laboratories.Dr. Potra has published 134 research papers in prestigious professional journals and over 30 papers in refereed proceedings. He is the Regional Editor for the Americas of the journal "Optimization Methods and Software", and serves on the editorial board of three other well known mathematical journals.