  A multidimensional optimization problem is formulated in the tropical
mathematics setting as to maximize a nonlinear objective function, which is
defined through a multiplicative conjugate transposition operator on vectors in
a finite-dimensional semimodule over a general idempotent semifield. The study
is motivated by problems drawn from project scheduling, where the deviation
between initiation or completion times of activities in a project is to be
maximized subject to various precedence constraints among the activities. To
solve the unconstrained problem, we first establish an upper bound for the
objective function, and then solve a system of vector equations to find all
vectors that yield the bound. As a corollary, an extension of the solution to
handle constrained problems is discussed. The results obtained are applied to
give complete direct solutions to the motivating problems from project
scheduling. Numerical examples of the development of optimal schedules are also
presented.
