A direct numerical method for quantitative pursuit-evasion games with state and control constraints is developed. The proposed method consists of update and correction procedures. In the update procedure, the evader`s control is perturbed using the gradient in the direction of maximizing the payoff and the pursuer`s control is determined to provide an intercept. The motion of the pursuer is optimized during the correction procedure. Through the update and the correction, the solution remains on the reaction curve of the pursuer and is perturbed toward a saddle-point equilibrium. State constraints are treated as the general payoff using the penalty function. The algorithm is applied to missile-aircraft engagements with state constraints. The horizontal engagement is subject to constraints on the look-angle of the missile seeker. The dynamic pressure of the aircraft is restricted in the vertical engagement.