Extremal properties of tropical eigenvalues and solutions to tropical optimization problems

Abstract An unconstrained optimization problem is formulated in terms of tropical mathematics to minimize a functional that is defined on a vector set by a matrix and calculated through multiplicative conjugate transposition. For some particular cases, the minimum in the problem is known to be equal to the tropical spectral radius of the matrix. We examine the problem in the common setting of a general idempotent semifield. A complete direct solution in a compact vector form is obtained to this problem under fairly general conditions. The result is extended to solve new tropical optimization problems with more general objective functions and inequality constraints. Applications to real-world problems that arise in project scheduling are presented. To illustrate the results obtained, numerical examples are also provided.

[1]  Nikolai Krivulin,et al.  A multidimensional tropical optimization problem with a non-linear objective function and linear constraints , 2013, ArXiv.

[2]  Marianne Akian,et al.  Max-Plus Algebra , 2006 .

[3]  N. Krivulin.  A complete closed-form solution to a tropical extremal problem , 2012, 1210.3658.

[4]  Nikolai K. Krivulin,et al.  A tropical extremal problem with nonlinear objective function and linear inequality constraints , 2012, ArXiv.

[5]  Michel Minoux,et al.  Graphs, dioids and semirings : new models and algorithms , 2008 .

[6]  B. Ciffler.  Scheduling general production systems using schedule algebra , 1963 .

[7]  S. N. N. Pandit,et al.  A New Matrix Calculus , 1961 .

[8]  Gernot M. Engel,et al.  Diagonal similarity and equivalence for matrices over groups with 0 , 1975 .

[9]  P. Butkovic.  Max-linear Systems: Theory and Algorithms , 2010 .

[10]  Geert Jan Olsder,et al.  Max Plus at Work-Modelling and Analysis of Synchronized Systems , 2006 .

[11]  J. Golan.  Semirings and Affine Equations over Them: Theory and Applications , 2003 .

[12]  Willi Hock,et al.  Lecture Notes in Economics and Mathematical Systems , 1981 .

[13]  Jean-Charles Billaut,et al.  Multicriteria scheduling , 2005, Eur. J. Oper. Res..

[14]  Nikolai Krivulin,et al.  A constrained tropical optimization problem: complete solution and application example , 2013, ArXiv.

[15]  N. Krivulin.  A new algebraic solution to multidimensional minimax location problems with Chebyshev distance , 2012, 1210.4770.

[16]  Nikolai K. Krivulin,et al.  Algebraic solutions to multidimensional minimax location problems with Chebyshev distance , 2011, ArXiv.

[17]  G. Litvinov.  Maslov dequantization, idempotent and tropical mathematics: A brief introduction , 2005, math/0507014.

[18]  Solution of generalized linear vector equations in idempotent algebra , 2006 .

[19]  Evaluation of bounds on the mean rate of growth of the state vector of a linear dynamical stochastic , 2005 .

[20]  Ludwig Elsner,et al.  Max-algebra and pairwise comparison matrices , 2004 .

[21]  Erik Demeulemeester,et al.  Project scheduling : a research handbook , 2002 .

[22]  B. Carré.  An Algebra for Network Routing Problems , 1971 .

[23]  R. A. Cuninghame-Green,et al.  Describing Industrial Processes with Interference and Approximating Their Steady-State Behaviour , 1962 .

[24]  N. Krivulin.  An extremal property of the eigenvalue of irreducible matrices in idempotent algebra and solution of the Rawls location problem , 2011 .