Microwave imaging is an emerging technique with applications in various areas, such as geoscience, aviation, and medicine. One of the classic methodologies to address this inverse problem is the Born iterative method (BIM), which divides the nonlinear problem into two linear subproblems: a forward and an inverse one. Several methods can be coupled within this framework to solve the inverse subproblem. This work proposes a formulation for the inverse subproblem in terms of a quadratic programming one, which allows the application of an exact methodology with an adequate mechanism to enforce a particular type of a priori information: the limits of the range of possible contrast values of the objects. This is one of the main contributions of this article and an adaption of the formulation to the use of multiple frequencies and regularization functions. The methodology was able to reconstruct images with few iterations. Several experiments were conducted to evaluate the performance of the proposed methodology in different scenarios considering synthesized geometries and breast phantoms. Mean error per pixel of less than 0.25% for synthesized geometries and 4.6% for breast phantoms (mean value of 0.74% among the experiments) were obtained for the approximations of the scattered field data and the contrast map, in addition to adequately reconstructing objects with different positions, geometries, and contrasts. The comparison between the original BIM and the proposed method with only Tikhonov regularization and no enforcement of prior information showed similar performance. On the other hand, the proposed method with the enforcement of prior information has presented a better performance than the original BIM in 14 of 19 experiments related to the synthesized geometries. The ability to enforce a priori information in the proposed method on the lower and upper bounds of the contrast was responsible for reducing the error by 25%, on average, with respect to the same obtained in the experiments when this information was not considered. In addition, the results also suggested that the enforcement of this information was able to speed up the convergence of the proposed method.