A constrained tropical optimization problem: complete solution and application example

A multidimensional optimization problem is considered, which is formulated in terms of tropical mathematics as to minimize a nonlinear objective function subject to linear inequality constraints. The optimization problem is motivated by 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. To solve the problem, we follow the approach based on the introduction of an additional unknown variable to reduce the problem to solving a linear inequality, where 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 general solution of the inequality is taken as a complete solution of the original problem. Under fairly general conditions, a complete direct solution to the problem is obtained in a compact vector form. The obtained result is applied to solve the motivating problem in project scheduling and illustrated with a numerical example. Key-Words: idempotent semifield, finite-dimensional semimodule, tropical optimization problem, nonlinear objective function, linear inequality constraint, project scheduling. MSC (2010): Primary 65K10; Secondary 15A80, 90C48, 90B35