Abstract: Idempotent algebra, which deals with vector semimodules over idempotent semirings, finds expanding application as a promising modeling and solution tool in applied mathematics, computer science, and operations research. The progress in the area is mainly due to the fact that many complicated problems that are actually nonlinear in the ordinary sense become linear and so more tractable when translated into the language of the algebra. Specifically, many classical problems in graph optimization and dynamic programming reduce to solving linear vector equations, finding eigenvalues and eigenvectors of matrices, and to similar computational procedures in the idempotent algebra setting. To illustrate, we examine both unconstrained and constrained multidimensional minimax single facility location problems with rectilinear and Chebyshev distances.
We begin with an overview of preliminary definitions and results in idempotent algebra, including basic concepts of scalar and matrix algebra, and elements of the spectral theory of matrices. A new algebraic approach based on investigation of extremal properties of eigenvalues for irreducible matrices is developed to solve multidimensional problems that involve minimization of functionals defined on idempotent vector semimodules. Furthermore, we provide a conventional description for the location problems of interest and then represent the problem in terms of idempotent algebra. An algebraic solution is given to unconstrained problems that reduce them to evaluating eigenvalues and eigenvectors of an appropriate matrix. The solution is subsequently extended to problems under constraints on the feasible location set. We conclude with a brief discussion about possible lines of further development and new area of applications of the algebraic approach.

Brief Biography of the Speaker:
Nikolai Krivulin received a university degree in applied mathematics and operations research in 1983 from St. Petersburg State University. He got his Ph.D. degree in 1990 and D.Sc. degree in 2010 both in applied mathematics from the same university. In 1983 he joined the Computer Center at St. Petersburg State University as a system software engineer, and in 1985 started his Ph.D. study. In 1987 he joined the Faculty of Mathematics and Mechanics at St. Petersburg State University as an Assistant Professor, and became an Assistant Professor there in 1990. From 1999 to 2002 he was the head of the Department of Information Management at the Graduate School of Management of the same university.
He is currently an Associate Professor of the Department of Statistical Modelling at St. Petersburg State University. His research interests include theory and applications of idempotent algebra, modelling and performance evaluation of queueing systems, methods of optimization, computational statistics and computer simulation. Nikolai Krivulin is an author and coauthor of more than 70 publications including papers published in reviewed journals and conference proceedings, books chapters, textbooks, and a monograph. He is a grantee of national and international foundations, including the Russian Foundation for Basic Research, the Russian Foundation for Humanities Research, the NATO Science Foundation, the USIA and Eurasia Foundation (USA), and the Royal Society (UK). He was a member of program and organizing committees of international conferences on mathematics, computer sciences, and information technology. He is a member of St. Petersburg Mathematical Society, AMS, and SIAM.