Abstract
Problem to date: Similar in nature to the knapsack problem and the indivisible investment problem, there exists a static optimal problem the variables of which have both discrete and continuous values as the optimum variables: that is, a mixed-integer programming problem as one optimal problem of the mixed programming problems. A relaxed method or a group method etc. has been used for these problems hitherto. Just like the dynamical indivisible investment problem and the resource allotment problem/the disposition of personnel problem through several periods, there are few optimal control methods for the mixed-quantized dynamical optimal control problem which has both discrete and continuous values. The proposed method in this paper: For the mixed-quantized optimal control problem in which the state equation is linear, the control problem is given by the formulation of Halkin’s discrete time optimal control problem. The mixed-quantized discrete maximum principle is given as the algorithm for this control problem. Where, for the maximization of Hamiltonian in each discrete time, the relaxed method—which improved branch and bound method—is used. Effects obtained in this paper: As the applications in this control problem, the above dynamical investment and the allotment problem through the several periods etc. are considered. In this paper, the solution to the discrete time mixed-quantized optimal control problem is given, and the efficiency of this method (mixed-quantized discrete maximum principle)—which is applied infrequently to this field—is shown, along with a numerical example.
