With the development of hyperspectral imaging technology and detection algorithm research, hyperspectral target detection has become indispensable in the field of target recognition and reconnaissance. But limited by the complex distribution of land covers and the difficulties to distinguish spectral characteristics between the target and its surroundings, existing target detection algorithms may still be insufficient on identifying target spectrum and suppressing background spectra. In this paper, we propose a novel iterative background reconstruction and suppression framework for hyperspectral target detection. By means of iteratively integrating the dictionary training algorithm with sparse-coding method to estimate the background training dictionary and its corresponding sparse representation, hyperspectral imagery (HSI) dataset is purified into background subspace matrix. Then, we add the constructed background suppression regularization term on the constrained energy minimization model to form the framework as a linearly background constrained optimization filter model. The optimal solution is then solved by a convex optimization interior-point method and improved by an iteratively reweighting method to achieve the goal of separating the target and the background. The proposed algorithm is applied to two synthetic hyperspectral datasets and two real-HSI datasets; the experimental results show that our algorithm performs a superior detection result compared to other classical hyperspectral target detection algorithms. It also provides a fresheye idea to integrate a variety of theoretical algorithms to explore the way of constructing and solving target detection model.