AbstractIn this article, the rank‐1 approximation of a nonnegative tensor is considered. Mathematically, the approximation problem can be formulated as an optimization problem. The Karush–Kuhn–Tucker (KKT) point of the optimization problem can be obtained by computing the nonnegative Z‐eigenvectoryof enlarged tensor . Therefore, we propose an iterative method with prediction and correction steps for computing nonnegative Z‐eigenvectoryof enlarged tensor , called the continuation method. In the theoretical part, we show that the computation requires only flops for each iteration and the computed Z‐eigenvectoryhas nonzero component block, and hence, the KKT point can be obtained. In addition, we show that the KKT point is a local optimizer of the optimization problem. Numerical experiments are provided to support the theoretical results.