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A multidimensional optimization problem is formulated in the tropicalmathematics setting as to maximize a nonlinear objective function, which isdefined through a multiplicative conjugate transposition operator on vectors ina finite-dimensional semimodule over a general idempotent semifield. The studyis motivated by problems drawn from project scheduling, where the deviationbetween initiation or completion times of activities in a project is to bemaximized subject to various precedence constraints among the activities. Tosolve the unconstrained problem, we first establish an upper bound for theobjective function, and then solve a system of vector equations to find allvectors that yield the bound. As a corollary, an extension of the solution tohandle constrained problems is discussed. The results obtained are applied togive complete direct solutions to the motivating problems from projectscheduling. Numerical examples of the development of optimal schedules are alsopresented.

A maximization problem in tropical mathematics: a complete solution and application examples