Special Issue "Idempotent Mathematics and Its Applications in Mathematical Physics and Mathematical Economy"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (28 February 2022) | Viewed by 1889

Special Issue Editors

Prof. Dr. Viktor Maslov
E-Mail Website
Guest Editor
Moscow Institute of Electronics and Mathematics, National Research University Higher School of Economics, 101000 Moscow, Russia
Interests: mathematics; thermodynamics; mathematical physics; mathematical modeling and optimization; differential equations; functional analysis and operator algebra; classical and quantum mechanics; nuclear physics
Prof. Dr. Vassili Kolokoltsov
E-Mail Website
Guest Editor
Department of Statistics, University of Warwick, Coventry CV4 7AL, UK
Interests: probability and stochastic processes; optimization and games with applications to business, biology and finances; mathematical physics; differential equations and functional analysis

Special Issue Information

Dear Colleagues,

Idempotent or tropical mathematics is the family of mathematical disciplines that use idempotent semirings and semifields instead of fields. This substitution of fields by idempotent semifields and semirings could be applied to many constructions from traditional mathematics. The language of idempotent mathematics (or idempotent analysis or idempotent calculus) creates a unified theory for treating a wide class of the optimization problems via linear (in the new algebra) methods leading theoretically to a circle of result that can be referred to as the idempotent functional analysis.

Idempotent or tropical mathematics is an asymptotic version of traditional mathematics. It can be considered as the result of the application of the dequantization procedure to traditional mathematics. The Planck constant in this asymptotic analysis belongs to the imaginary axis and goes to zero. New asymptotic methods are developed based on this idea.

Methods of idempotent analysis are useful for numerical calculations of the systems of differential or algebraic equations often leading to curse-of-dimensionality-free methods.

The numeric methods provide the main asymptotic terms with interval estimations of solutions. Application of idempotent mathematics are numerous in mathematical physics (de-quantization, thermodynamics), chemical engineering and system biology (including chemical kinetics and the analysis of multiscale dynamics), the theory of dimension and entropy (including fractal and negative dimensions), in models of economics and finances, in the theory of complexity, approximation theory in high dimensions, and nonlinear problems of optimal control and optimization, among others.

Prof. Dr. Viktor Maslov
Prof. Dr. Vassili Kolokoltsov
Guest Editors

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Keywords

  • Idempotent algebra
  • Tropical mathematics
  • Discrete event dynamic system
  • Queueing networks
  • Computer simulation
  • Deterministic and stochastic optimal control problems by the dynamic programming method
  • Numerical analysis and algorithms
  • Max-plus or tropical algebras
  • Idempotent measures and large deviations
  • Perturbations of eigenvalues
  • Monotone nonexpansive maps and nonlinear Perron–Frobenius theory

Published Papers (2 papers)

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Research

Article
Algebraic Solution of Tropical Polynomial Optimization Problems
Mathematics 2021, 9(19), 2472; https://doi.org/10.3390/math9192472 - 03 Oct 2021
Viewed by 601
Abstract
We consider constrained optimization problems defined in the tropical algebra setting on a linearly ordered, algebraically complete (radicable) idempotent semifield (a semiring with idempotent addition and invertible multiplication). The problems are to minimize the objective functions given by tropical analogues of multivariate Puiseux [...] Read more.
We consider constrained optimization problems defined in the tropical algebra setting on a linearly ordered, algebraically complete (radicable) idempotent semifield (a semiring with idempotent addition and invertible multiplication). The problems are to minimize the objective functions given by tropical analogues of multivariate Puiseux polynomials, subject to box constraints on the variables. A technique for variable elimination is presented that converts the original optimization problem to a new one in which one variable is removed and the box constraint for this variable is modified. The novel approach may be thought of as an extension of the Fourier–Motzkin elimination method for systems of linear inequalities in ordered fields to the issue of polynomial optimization in ordered tropical semifields. We use this technique to develop a procedure to solve the problem in a finite number of iterations. The procedure includes two phases: backward elimination and forward substitution of variables. We describe the main steps of the procedure, discuss its computational complexity and present numerical examples. Full article
Article
Algebraic Solution to Constrained Bi-Criteria Decision Problem of Rating Alternatives through Pairwise Comparisons
Mathematics 2021, 9(4), 303; https://doi.org/10.3390/math9040303 - 04 Feb 2021
Viewed by 662
Abstract
We consider a decision-making problem to evaluate absolute ratings of alternatives from the results of their pairwise comparisons according to two criteria, subject to constraints on the ratings. We formulate the problem as a bi-objective optimization problem of constrained matrix approximation in the [...] Read more.
We consider a decision-making problem to evaluate absolute ratings of alternatives from the results of their pairwise comparisons according to two criteria, subject to constraints on the ratings. We formulate the problem as a bi-objective optimization problem of constrained matrix approximation in the Chebyshev sense in logarithmic scale. The problem is to approximate the pairwise comparison matrices for each criterion simultaneously by a common consistent matrix of unit rank, which determines the vector of ratings. We represent and solve the optimization problem in the framework of tropical (idempotent) algebra, which deals with the theory and applications of idempotent semirings and semifields. The solution involves the introduction of two parameters that represent the minimum values of approximation error for each matrix and thereby describe the Pareto frontier for the bi-objective problem. The optimization problem then reduces to a parametrized vector inequality. The necessary and sufficient conditions for solutions of the inequality serve to derive the Pareto frontier for the problem. All solutions of the inequality, which correspond to the Pareto frontier, are taken as a complete Pareto-optimal solution to the problem. We apply these results to the decision problem of interest and present illustrative examples. Full article
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