Many algorithms from digital signal processing, including digital filters or discrete transforms, require the multiplications with several constants. These can be efficiently implemented multiplierless by using additions, subtractions, and bit-shifts. Finding a multiplierless solution with minimal cost is known as the multiple constant multiplication (MCM) problem. Usually, not the full precision is required at the output. The state-of-the-art approaches consist in finding an MCM solution first, and truncating it in a second step. In this work, we solve the MCM problem with minimal number of full adders for truncated outputs. By combining the two steps into a global optimization problem, modeled through mixed-integer linear programming, we are able to reduce the number of full adders by 60% in best cases and by 12% on average. Our method has shown its efficiency on more than 80 instances from literature and permits a fast improvement of state-of-the-art results in most of the cases.