Abstract: We consider the quadratic semi-assignment problem in which we minimize a quadratic pseudo-Boolean function F subject to the semi-assignment constraints. We propose in this paper a linear programming method to obtain the best reduction of this problem, i.e. to compute the greatest constant c such that F is equal to c plus F' for all feasible solutions, F' being a quadratic pseudo-Boolean function with nonnegative coefficients. Thus constant c can be viewed as a generalization of the height of an unconstrained quadratic 0-1 function introduced by Hammer, Hansen and Simeone, Roof duality, complementation and persistency in quadratic 0-1 optimization (Math. Programming, 28 (1984), pp. 121-195) to constrained quadratic 0-1 optimization. Finally, computational experiments proving the practical usefulness of this reduction are reported.