A multidimensional tropical optimization problem with a non-linear objective function and linear constraints
A multidimensional tropical optimization problem with a non-linear objective function and linear constraints
复制标题
DOI：
10.1080/02331934.2013.840624
复制
发表时间：
2013-03
期刊：
影响因子：
2.2
通讯作者：
N. Krivulin
中科院分区：
文献类型：
--
作者：
N. Krivulin
我们研究了一个多维优化问题的热带数学设置。该问题涉及定义在幂等半域上的有限维半模上的非线性函数在线性不等式约束下的最小化。我们首先概述了已知的热带优化问题的线性和非线性目标函数。一个简短的介绍热带代数提供了一个正式的框架，解决正在研究的问题。作为初步结果，给出了一个含有任意矩阵的线性不等式的解法。我们描述了一个例子，从项目调度的优化问题，然后提供了一个一般表示的问题。为了解决这个问题，我们引入了一个额外的变量，并减少了问题的线性不等式的解决，其中变量的作用是一个参数。不等式成立的一个充分必要条件被用来计算参数，而不等式的解被认为是问题的解。基于这种方法，在相当一般的条件下，一个完整的直接解决方案，在一个紧凑的向量形式的最优化问题。二维问题的数值和图形的例子来说明所得到的结果。
We examine a multidimensional optimization problem in the tropical mathematics setting. The problem involves the minimization of a non-linear function defined on a finite-dimensional semimodule over an idempotent semifield subject to linear inequality constraints. We start with an overview of known tropical optimization problems with linear and non-linear objective functions. A short introduction to tropical algebra is provided to offer a formal framework for solving the problem under study. As a preliminary result, a solution to a linear inequality with an arbitrary matrix is presented. We describe an example optimization problem drawn from project scheduling and then offer a general representation of the problem. To solve the problem, we introduce an additional variable and reduce the problem to the solving of a linear inequality, in which the variable plays the role of a parameter. A necessary and sufficient condition for the inequality to hold is used to evaluate the parameter, whereas the solution to the inequality is considered a solution to the problem. Based on this approach, a complete direct solution in a compact vector form is derived for the optimization problem under fairly general conditions. Numerical and graphical examples for two-dimensional problems are given to illustrate the obtained results.