We propose a new linear model-predictive control scheme in multi-contact based on the center of mass reduced model. In order to linearize the dynamics of the CoM, we exploit the notion of stability polyhedrons associated to given contact stances, inside which the existence of contact forces is guaranteed. To compute stability polyhedrons, we have first to specify a convex polytope inside which the center of mass's acceleration lies. We then generate a minimum jerk trajectory inside these successive stability polyhedrons that also yields contact transitions timings, and integrate it with our quadratic programming whole-body controller as part of the multi-objective tasks.