For the nonlinear acoustic echo cancellation, we present an algorithm to estimate the threshold of the clipping effect and the room impulse response vector by suppressing their time-varying cost function. A common way to suppress the time-varying cost function of a pair of parameters is to alternatingly minimize the function with respect to each parameter while keeping the other fixed, which we refer to as adaptive alternating minimization. However, since the cost function for the threshold is nonconvex, the conventional methods approximate the exact minimizations by gradient descent updates, which causes serious degradation of the estimation accuracy in some occasions. In this paper, by exploring the fact that the cost function for the threshold becomes piecewise quadratic, we propose to exactly minimize the cost function for the threshold in a closed form while suppressing the cost function for the impulse response vector in an online manner, which we call exact-online adaptive alternating minimization. The proposed method is expected to approximate more efficiently the adaptive alternating minimization strategy than the conventional methods. Numerical experiments demonstrate the efficacy of the proposed method.