Minimax Algebra

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A number of different problems of interest to the operational researcher and the mathematical economist - for example, certain problems of optimization on graphs and networks, of machine-scheduling, of convex analysis and of approx imation theory - can be formulated in a convenient way using the algebraic structure (R,$,@) where we may think of R as the (extended) real-number system with the binary combining operations x$y, x ®y defined to be max(x,y),(x+y) respectively. The use of this algebraic structure gives these problems the character of problems of linear algebra, or linear operator theory. This fact hB.s been independently discovered by a number of people working in various fields and in different notations, and the starting-point for the present Lecture Notes was the writer’s persuasion that the time had arrived to present a unified account of the algebra of linear transformations of spaces of n-tuples over (R,$, ®),to demonstrate its relevance to operational research and to give solutions to the standard linear-algebraic problems which arise - e.g. the solution of linear equations exactly or approximately, the eigenvector eigenvalue problem andso on.Some of this material contains results of hitherto unpublished research carried out by the writer during the years 1970-1977.