We are concerned herein with inverse scattering problems in stratified media and aspect-limited data configurations. In such configurations, the sources and receivers of the probing waves are located in a medium different from the one which contains the object under test. This results in a lack of information which enhances the inherent ill-posedness of the inverse problem. To make the problem more tractable, we assume that the test object is homogeneous with known constitutive parameters so that the inverse problem consists of reconstructing its shape and location. This non-linear inverse problem is solved using the modified gradient method in which the a priori information is introduced as a binary constraint. A cooling parameter is introduced at the same time, which allows us to control the evolution of the iterative process. The effectiveness of this algorithm is studied for three different physical applications.