Direct solutions to tropical optimization problems with nonlinear objective functions and boundary constraints

We examine two multidimensional optimization problems that are formulated in terms of tropical mathematics. The problems are to minimize nonlinear objective functions, which are defined through the multiplicative conjugate vector transposition on vectors of a finite-dimensional semimodule over an idempotent semifield, and subject to boundary constraints. The solution approach is implemented, which involves the derivation of the sharp bounds on the objective functions, followed by determination of vectors that yield the bound. Based on the approach, direct solutions to the problems are obtained in a compact vector form. To illustrate, we apply the results to solving constrained Chebyshev approximation and location problems, and give numerical examples.Comment: Mathematical Methods and Optimization Techniques in Engineering: Proc. 1st Intern. Conf. on Optimization Techniques in Engineering (OTENG '13), Antalya, Turkey, October 8-10, 2013, WSEAS Press, 2013, pp. 86-91. ISBN 978-960-474-339-

Complete solution of a constrained tropical optimization problem with application to location analysis

We present a multidimensional optimization problem that is formulated and solved in the tropical mathematics setting. The problem consists of minimizing a nonlinear objective function defined on vectors over an idempotent semifield by means of a conjugate transposition operator, subject to constraints in the form of linear vector inequalities. A complete direct solution to the problem under fairly general assumptions is given in a compact vector form suitable for both further analysis and practical implementation. We apply the result to solve a multidimensional minimax single facility location problem with Chebyshev distance and with inequality constraints imposed on the feasible location area.Comment: 20 pages, 3 figure

A constrained tropical optimization problem: complete solution and application example

The paper focuses on a multidimensional optimization problem, which is formulated in terms of tropical mathematics and consists in minimizing a nonlinear objective function subject to linear inequality constraints. To solve the problem, we follow an approach based on the introduction of an additional unknown variable to reduce the problem to solving linear inequalities, where the variable plays the role of a parameter. A necessary and sufficient condition for the inequalities to hold is used to evaluate the parameter, whereas the general solution of the inequalities is taken as a solution of the original problem. Under fairly general assumptions, a complete direct solution to the problem is obtained in a compact vector form. The result is applied to solve a problem in project scheduling when an optimal schedule is given by minimizing the flow time of activities in a project under various activity precedence constraints. As an illustration, a numerical example of optimal scheduling is also presented.Comment: 20 pages, accepted for publication in Contemporary Mathematic