This paper addresses the problem of static output-feedback control design for uncertain discrete-time Lur'e systems with sector and slope bounded nonlinearities. By manipulating stability analysis conditions from the literature based on integralquadratic-constraint theory and multipliers, and applying Finsler's lemma, new synthesis conditions are provided in terms of sufficient linear matrix inequalities. After applying a relaxation in the stability conditions, an iterative algorithm is proposed to search for the two stabilizing gains of the control law, one for the output of the plant and one for the output of the nonlinear branch. The gains, that appear affinely in the inequalities, are treated as variables of the optimization problem. Therefore, the approach can handle state or output-feedback indistinctly and cope with magnitude or structural constraints (such as decentralization) in the gains without introducing additional conservatism. Numerical examples illustrate the results.