Abstract
We present a new, simple compact proof of the known strong duality theorem of tropical linear programming with one-sided constraints. This result together with properties of subeigenvectors enables us to directly solve a special tropical linear program with two-sided constraints. We also study the duality gap in tropical integer linear programming. A direct solution is available for the primal problem. An algorithm of quadratic complexity is presented for the dual problem. A direct solution is available provided that all coefficients of the objective function are integer. This solution provides a good estimate of the optimal objective function value in the general case.
Highlights
IntroductionTropical linear algebra ( called max-algebra or path algebra) is an analogue of linear algebra, where addition is replaced by maximization and multiplication by conventional addition
Tropical linear algebra is an analogue of linear algebra, where addition is replaced by maximization and multiplication by conventional addition.The interest in tropical linear algebra was originally motivated by the possibility of dealing with a class of nonlinear problems in pure and applied mathematics, operational research, science and engineering as if they were linear due to the fact that the underlying structure is a commutative and idempotent semifield
We have presented a simple proof of the known strong duality theorem of tropical linear programming with one-sided constraints
Summary
Tropical linear algebra ( called max-algebra or path algebra) is an analogue of linear algebra, where addition is replaced by maximization and multiplication by conventional addition. The tropical eigenvalue–eigenvector problem (briefly eigenproblem) is the following: Given A ∈ Rn×n, find all λ ∈ R (eigenvalues) and x ∈ Rn, x = ε (eigenvectors) such that This problem has been studied since the work of Cuninghame-Green. We will usually not write the operator ⊗ , and for matrices, the convention applies that if no operator appears, the product is in min-algebra whenever it follows the symbol #, otherwise it is in max-algebra. In this way, a residuated pair of operations (a special case of the Galois connection) has been defined, namely.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
