Solution of a Multidimensional Tropical Optimization Problem Using Matrix Sparsification
A complete solution is proposed for the problem of minimizing a function defined on vectors with elements in a tropical (idempotent) semifield. The tropical optimization problem under consideration arises, for example, when we need to find the best (in the sense of the Chebyshev metric) approximate solution to tropical vector equations and occurs in various applications, including scheduling, location, and decision-making problems. To solve the problem, the minimum value of the objective function is determined, the set of solutions is described by a system of inequalities, and one of the solutions is obtained. Thereafter, an extended set of solutions is constructed using the sparsification of the matrix of the problem, and then a complete solution in the form of a family of subsets is derived. Procedures that make it possible to reduce the number of subsets to be examined when constructing the complete solution are described. It is shown how the complete solution can be represented parametrically in a compact vector form. The solution obtained in this study generalizes known results, which are commonly reduced to deriving one solution and do not allow us to find the entire solution set. To illustrate the main results of the work, an example of numerically solving the problem in the set of three-dimensional vectors is given.
Keywordsidempotent semifield tropical optimization Chebyshev approximation complete solution matrix sparsification
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- 2.V. P. Maslov and V. N. Kolokoltsov, Idempotent Analysis and Its Applications to Optimal Control Theory (Nauka, Moscow, 1994) [in Russian].Google Scholar
- 4.J. S. Golan, Semirings and Affine Equations over Them: Theory and Applications (Springer-Verlag, Dordrecht, 2003), in Ser.: Mathematics and Its Applications, Vol. 556. https://doi.org/10.1007/978-94-017-0383-3.Google Scholar
- 6.M. Gondran and M. Minoux, Graphs, Dioids and Semirings: New Models and Algorithms (Springer-Verlag, New York, 2008), in Ser.: Operations Research / Computer Science Interfaces, Vol. 41. https://doi.org/10.1007/978-0-387-75450-5.Google Scholar
- 7.N. K. Krivulin, Methods of Idempotent Algebra for Problems in Modeling and Analysis of Complex Systems (S.-Peterb. Gos. Univ., St. Petersburg, 2009) [in Russian].Google Scholar
- 11.K. Zimmermann, “Some optimization problems with extremal operations,” in Mathematical Programming at Oberwolfach II, Ed. by B. Korte and K. Ritter (Springer-Verlag, Berlin, 1984), in Ser.: Mathematical Programming Studies, Vol. 22, pp. 237–251. https://doi.org/10.1007/BFb0121020.MathSciNetCrossRefzbMATHGoogle Scholar
- 13.P. Butkovic and K. P. Tam, “On some properties of the image set of a max-linear mapping,” in Tropical and Idempotent Mathematics, Ed. by G. L. Litvinov, S. N. Sergeev (AMS, Providence, RI, 2009), in Ser.: Contemporary Mathematics, Vol. 495, pp. 115–126. https://doi.org/10.1090/conm/495/09694.MathSciNetCrossRefzbMATHGoogle Scholar
- 14.N. K. Krivulin, “On solution of linear vector equations in idempotent algebra,” in Mathematical Models. Theory and Applications, Ed. by M. K. Chirkov (VVM, St. Petersburg, 2004), Vol. 5, pp. 105–113 [in Russian].Google Scholar
- 15.N. Krivulin, “A new algebraic solution to multidimensional minimax location problems with Chebyshev distance,” WSEAS Trans. Math. 11, 605–614 (2012).Google Scholar
- 16.N. Krivulin, “Solution of linear equations and inequalities in idempotent vector spaces,” Int. J. Appl. Math. Inform. 7, 14–23 (2013).Google Scholar
- 17.N. Krivulin and K. Zimmermann, “Direct solutions to tropical optimization problems with nonlinear objective functions and boundary constraints,” in Mathematical Methods and Optimization Techniques in Engineering, Ed. by by D. Biolek, H. Walter, I. Utu, and C. von Lucken (WSEAS, 2013), pp. 86–91.Google Scholar