Projections in minimax algebra

Abstract

An axiomatic theory of linear operators can be constructed for abstract spaces defined over (R, *, e), that is over the (extended) real numbers R with the binary operations x + y = max (x,y) and x ~ y = x + y. Many of the features of conventional linear operator theory can be reproduced in this theory, although the proof techniques are quite different. Specialisation of the theory to spaces of n-tuples provides techniques for analysing a number of well-known operational research problems, whilst specialisation to function spaces provides a natural formal framework for certain familiar problems of approximation, optimisation and duality. A motivating exampleThe object of the present article is to give an overview of a system of ideas which, because of the extensive nature of the research program of which they form a part (and the long refereeing procedures so implied), will not all be published in detailed form for some time. The accent in what follows is on explanation rather than on proof.In the past 20 years (see references at the end) several different authors, often apparently unaware of one another's work, have discovered that an attractive formulation language is provided for a surprisingly wide class of problems by setting up an algebra of real numbers (perhaps extended by symbols such as -~, etc.) in which, however, the usual operations of multiplication and addition of two numbers are replaced by the operations:(i) arithmetical addition, and (ii) selection of the greater (or dually, the smaller) of the two numbers, respectively. Thus, if R is the set of (suitably extended) real numbers, we define for 111

A multidimensional tropical optimization problem with a non-linear objective function and linear constraints

“…Multidimensional optimisation problems form a significant research domain in tropical mathematics, dating back to the works [23,12]. These problems arise in real-world applications in various areas.…”

“…These problems arise in real-world applications in various areas. Specifically, [12,48,44,47,6,7,2] provide tropical solutions to problems in project scheduling. Fur-thermore, [13,25,33,27] solve problems in location analysis.…”

“…A minimax earliness problem in machine scheduling that was previously studied [12] is an early instance of the real-world problems that can be represented and solved within the framework of tropical mathematics. Further examples include multidimensional optimisation problems in location analysis, transportation networks, decision making, and discrete event systems.…”

“…In another important class of problems that dates back in the literature to the works [12,10,48], the constraints are linear, but the objective function is not. This class actually contains problems with functions that involve a conjugate transposition operator, which does not preserve tropical linearity.…”

“…Some problems, such as those in the early works [23,12,40,48], can be completely solved, and the solution is obtained in an explicit vector form in terms of a general semiring. In contrast, the existing solutions given for other problems concentrate on a particular semifield and are obtained in the form of iterative algorithms that give particular solutions (if there are any) or indicate that there is no solution (see, for instance, [4,44]).…”