# Solution of a tropical optimization problem using matrix sparsification

Research output

### 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 [1-3]. 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
 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. 163-177.
 Krivulin, N. A multidimensional tropical optimization problem with nonlinear objective function and linear constraints. Optimization, vol. 64, no. 5, pp. 1107-1129, 2015.
 Krivulin, N. Extremal properties of tropical eigenvalues and solutions to tropical optimization problems. Linear Algebra Appl., vol. 468, pp. 211-232, 2015.
Original language English 10-10 Published - Aug 2015 4th International Conference on Matrix Methods in Mathematics and Applications - Skolkovo Institute of Science and Technology, MoscowDuration: 24 Aug 2015 → 28 Aug 2015http://matrix.inm.ras.ru/program_and_abstracts.pdf

### Conference

Conference 4th International Conference on Matrix Methods in Mathematics and Applications MMMA-2015 Russian Federation Moscow 24/08/15 → 28/08/15 http://matrix.inm.ras.ru/program_and_abstracts.pdf

### Fingerprint

Idempotent
Optimization Problem
Nonlinear Function
Subset
Objective function
Semifield
Partial
Scalar multiplication
Seminorm
Optimization
Transposition
Backtracking
Linear Constraints
Threshold Value
Solution Set
Linear algebra
Multiplicative
Eigenvalue
Minimise

### Scopus subject areas

• Algebra and Number Theory

### Cite this

Кривулин, Н. К. (2015). Solution of a tropical optimization problem using matrix sparsification. 10-10. Abstract from 4th International Conference on Matrix Methods in Mathematics and Applications, Moscow, .
Кривулин, Николай Кимович. / Solution of a tropical optimization problem using matrix sparsification. Abstract from 4th International Conference on Matrix Methods in Mathematics and Applications, Moscow, .
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2015. 10-10 Abstract from 4th International Conference on Matrix Methods in Mathematics and Applications, Moscow, .

Research output

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N2 - 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 [1-3]. 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 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. 163-177. Krivulin, N. A multidimensional tropical optimization problem with nonlinear objective function and linear constraints. Optimization, vol. 64, no. 5, pp. 1107-1129, 2015. Krivulin, N. Extremal properties of tropical eigenvalues and solutions to tropical optimization problems. Linear Algebra Appl., vol. 468, pp. 211-232, 2015.

AB - 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 [1-3]. 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 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. 163-177. Krivulin, N. A multidimensional tropical optimization problem with nonlinear objective function and linear constraints. Optimization, vol. 64, no. 5, pp. 1107-1129, 2015. Krivulin, N. Extremal properties of tropical eigenvalues and solutions to tropical optimization problems. Linear Algebra Appl., vol. 468, pp. 211-232, 2015.

M3 - Abstract

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Кривулин НК. Solution of a tropical optimization problem using matrix sparsification. 2015. Abstract from 4th International Conference on Matrix Methods in Mathematics and Applications, Moscow, .