Technical Note—An Integer Maximization Problem

This note solves the following problem: For v_i > 0 and 0 < d < 1, maximize V(m₁, …, m_n) = ∑_i=1ⁱ⁼ⁿv_i(1 − d^m_i) with respect to (m₁, …, m_n), subject to the conditions that the m_i are nonnegative integers and ∑_i=1ⁱ⁼ⁿm_i = M (an integer). The solution is obtained by a simple rounding procedure applied to the (well known) solution of the corresponding continuous problem.