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Abstract

Consider the set A = R ∪ {+∞} with the binary operations ⊕ = max and = + and denote by A n the set of vectors v = (v 1 , ..., v n ) with entries in A. Let the generalised sum u ⊕ v of two vectors denote the vector with entries u j ⊕ v j , and the product a v of an element a ∈ A and a vector v ∈ A n denote the vector with the entries a vj. With these operations, the set A n provides the simplest example of an idempotent semimodule. The study of idempotent semimodules and their morphisms is the subject of idempotent linear algebra, which has been developing for about 40 years already as a useful tool in a number of problems of discrete optimisation. Idempotent analysis studies infinite dimensional idempotent semimodules and is aimed at the applications to the optimisations problems with general (not necessarily finite) state spaces. We review here the main facts of idempotent analysis and its major areas of applications in optimisation theory, namely in multicriteria optimisation, in turnpike theory and mathematical economics, in the theory of generalised solutions of the Hamilton-Jacobi Bellman (HJB) equation, in the theory of games and controlled Marcov processes, in financial mathematics.

Algebraic solutions of tropical optimization problems

“…The literature on the topic offers a wide range of publications written in varied styles and available on both the introductory and advanced levels. Specifically, the recent monographs: Kolokoltsov and Maslov [22], Golan [27], Heidergott, Olsder and van der Woude [28], Gondran and Minoux [29,31], Butkovič [31]; and overview papers: Kolokoltsov [40], Litvinov [41], Akian, Bapat and Gaubert [42], provide a broad source for further reading.…”