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Algebraic Solution of Tropical Polynomial Optimization Problems

Faculty of Mthematics and Mechanics, St. Petersburg State University, Universitetskaya Emb. 7/9, 199034 St. Petersburg, Russia
Academic Editors: Viktor Maslov and Vassili Kolokoltsov
Mathematics 2021, 9(19), 2472; https://doi.org/10.3390/math9192472
Received: 31 August 2021 / Revised: 22 September 2021 / Accepted: 25 September 2021 / Published: 3 October 2021
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. View Full-Text
Keywords: tropical algebra; idempotent semifield; tropical Puiseux polynomial; constrained polynomial optimization problem; box constraint; variable elimination tropical algebra; idempotent semifield; tropical Puiseux polynomial; constrained polynomial optimization problem; box constraint; variable elimination
MDPI and ACS Style

Krivulin, N. Algebraic Solution of Tropical Polynomial Optimization Problems. Mathematics 2021, 9, 2472. https://doi.org/10.3390/math9192472

AMA Style

Krivulin N. Algebraic Solution of Tropical Polynomial Optimization Problems. Mathematics. 2021; 9(19):2472. https://doi.org/10.3390/math9192472

Chicago/Turabian Style

Krivulin, Nikolai. 2021. "Algebraic Solution of Tropical Polynomial Optimization Problems" Mathematics 9, no. 19: 2472. https://doi.org/10.3390/math9192472

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