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Abstract
A multidimensional optimization problem, which arises in various applications in the form of minimization of span seminorm is considered in the framework of tropical (idempotent) mathematics. The problem is formulated to minimize a nonlinear function, which is defined on vectors over an idempotent semifield, given by a matrix, and calculated using multiplicative conjugate transposition. To solve the problem, we apply and develop methods of tropical optimization proposed and investigated in [13]. First, we find the minimum value of the objective function, and give a partial solution as a subset of vectors represented in an explicit form. We characterize all solutions to the problem by a system of simultaneous vector equation and inequality, and use this characterization to investigate properties of the solution set, which, in particular, turns out to be closed under vector addition and scalar multiplication. Furthermore, a matrix sparsification technique is developed to drop, without affecting the solution of the problem, those entries in the matrix which are below prescribed threshold values. By combining this technique with the above characterization, the previous partial solution is extended to a wider solution subset, and then to a complete solution described as a family of subsets. Finally, we offer a backtracking procedure that generates all members of the family, and derive an explicit representation for the complete solution in a compact vector form.
References
[1] Krivulin, N. A constrained tropical optimization problem: Complete solution and application example. In G. L. Litvinov and S. N. Sergeev (eds.) Tropical and Idempotent Mathematics and Applications, Contemp. Math. , vol. 616, Providence, RI: AMS, 2014, pp. 163177.
[2] Krivulin, N. A multidimensional tropical optimization problem with nonlinear objective function and linear constraints. Optimization, vol. 64, no. 5, pp. 11071129, 2015.
[3] Krivulin, N. Extremal properties of tropical eigenvalues and solutions to tropical optimization problems. Linear Algebra Appl., vol. 468, pp. 211232, 2015.
References
[1] Krivulin, N. A constrained tropical optimization problem: Complete solution and application example. In G. L. Litvinov and S. N. Sergeev (eds.) Tropical and Idempotent Mathematics and Applications, Contemp. Math. , vol. 616, Providence, RI: AMS, 2014, pp. 163177.
[2] Krivulin, N. A multidimensional tropical optimization problem with nonlinear objective function and linear constraints. Optimization, vol. 64, no. 5, pp. 11071129, 2015.
[3] Krivulin, N. Extremal properties of tropical eigenvalues and solutions to tropical optimization problems. Linear Algebra Appl., vol. 468, pp. 211232, 2015.
Original language  English 

Pages  10 
State  Published  Aug 2015 
Event  4th International Conference on Matrix Methods in Mathematics and Applications  Skolkovo Institute of Science and Technology, Moscow, Russian Federation Duration: 24 Aug 2015 → 28 Aug 2015 http://matrix.inm.ras.ru/program_and_abstracts.pdf 
Conference
Conference  4th International Conference on Matrix Methods in Mathematics and Applications 

Abbreviated title  MMMA2015 
Country/Territory  Russian Federation 
City  Moscow 
Period  24/08/15 → 28/08/15 
Internet address 
Scopus subject areas
 Algebra and Number Theory
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Solution of a tropical optimization problem using matrix sparsification
Николай Кимович Кривулин (Invited speaker)
24 Aug 2015Activity: Talk types › Invited talk

4th International Conference on Matrix Methods in Mathematics and Applications
Николай Кимович Кривулин (Participant)
24 Aug 2015 → 28 Aug 2015Activity: Attendance types › Participating in a conference, workshop, ...