The mixed-integer nonlinear bilevel programming problem can be described as a leader-follower game, in which the leader controls a set of variables in order to minimize a given objective function. Each choice of the leader is followed by a choice of the follower who minimizes a convex quadratic function over a polyhedral feasible set. A branch and bound algorithm is suggested for solving the cases when the leader's objective is a convex function of mixed-integer variables. In fact, the algorithm looks for a Kuhn-Tucker point of a follower's optimization problem which minimizes the leader's objective. Extensive numerical tests are conducted for choosing the most promising branching strategy.