To improve the poor infrared image quality that caused by stripe non-uniformity, a new robust iterative method is proposed for single image stripe non-uniformity correction. The proposed method is designed from following aspects. First, an analytical closed-form solution is derived for general non-uniformity correction problems, which builds a theoretical foundation and a better mathematical understanding of non-uniformity correction problems. Second, an optimization framework is developed through combination of closed-form solution and iterative method, which can result in a robust solution to stripe non-uniformity correction problem. Third, a new relaxed constraint is proposed to form a larger solution space and leads to a better global optimal solution. Furthermore, compared with other state-of-art methods, the proposed method can eliminate more stripe non-uniformity as well as achieve higher quality of the corrected images. Statistical numerical experiments are presented to demonstrate high robustness of this iterative method.