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Abstract
We consider multidimensional optimization problems, formulated in the framework of tropical (idempotent) mathematics to minimize or maximize functions defined on vectors over idempotent semifields. The objective functions can be linear or nonlinear in the tropical mathematics sense; the problems can have constraints in the form of tropical linear vector equalities and inequalities. We start with a brief overview of known optimization problems and existing solution methods. Some of the problems are solved directly in an explicit form under fairly general assumptions, whereas other problems have only algorithmic solutions in the form of iterative computational procedures, which produce a particular solution, or indicate that no solution exist. Furthermore, we examine new unconstrained and constrained problems with nonlinear objective functions, which are defined using multiplicative conjugate transposition of vectors. Examples include problems of Chebyshev approximation, problems of minimizing the span seminorm, and problems with evaluating the tropical spectral radius of a matrix. To solve the problems, we propose new techniques based on the reduction of the problem to parametrized systems of inequalities, the derivation sharp bounds for the objective function, and the application of extremal properties of the spectral radius. By using these techniques, we offer direct exact solutions of the problems in a compact vector form, which is ready for further analysis and practical implementation. For some problems, the solutions obtained are complete solutions. Finally, applications of the results to problems in Chebyshev approximation, project scheduling, location analysis and decision making are discussed.
Original language  English 

Pages  616 
State  Published  Jul 2016 
Event  7th European Congress of Mathematics  Technische Universität Berlin, Berlin, Germany Duration: 18 Jul 2016 → 22 Jul 2016 http://7ecm.de/ 
Conference
Conference  7th European Congress of Mathematics 

Abbreviated title  7ECM 
Country  Germany 
City  Berlin 
Period  18/07/16 → 22/07/16 
Internet address 
Scopus subject areas
 Control and Optimization
 Algebra and Number Theory
 Management Science and Operations Research
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Activities

7th European Congress of Mathematics
Николай Кимович Кривулин (Participant)
18 Jul 2016 → 22 Jul 2016Activity: Attendance types › Participating in a conference, workshop, ...

Optimization problems over tropical semifields: algebraic solutions and application examples
Николай Кимович Кривулин (Speaker)
20 Jul 2016Activity: Talk types › Oral presentation