Various effective algorithms have been proposed in the past two decades for nonlinear PDEs arising from the unconstrained total-variation-based image denoising problem regularizing the total variation constrained minimization model. Such algorithms can be used to obtain a satisfactory result as long as a suitable regularization parameter balancing the trade-off between a good fit to the data and a regular solution is given. However, it is generally difficult to obtain a suitable regularization parameter without which restored images can be unsatisfactory: if it is too large, then the resulting solution is still contaminated by noise, while if too small, the solution is a poor approximation of the true noise-free solution. To provide an automatic method for the regularization parameter when the noise level is known a priori, one way is to address the coupled Karush–Kuhn–Tucker (KKT) systems from the constrained total variation optimization problem. So far much less work has been done on this problem. This paper presents an iterative update algorithm for a Lagrange multiplier to solve the KKT conditions, and our proposed method can adaptively deal with noisy images with different variances $\sigma^2$. Numerical experiments show that our model can effectively find a highly accurate solution and produce excellent restoration results in terms of image quality.