This 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.