In this paper, we will develop novel formalism for the generalized linear least-squares inversion with arbitrary a priori information incorporated in the inversion in terms of ''soft'' and ''hard'' bounds. The technique can be a valuable tool for solving different problems, including those related to global seismology and high-resolution crustal imagery, especially when numerical algorithms break down due to the absence of properly chosen constraints. Due to different undesirable events in the recorded data (such as leg jumps, skips, significant noise level) the conventional least-squares in realistic situations can give erroneous solutions since the model parameters can easily readjust to variations in the measured data and give inaccurate or even non-physical results. Constrained inversion has been overlooked by geophysicists since the mathematics involved are much more complicated. We show that using the concept of duality, constrained problems can be transformed to unconstrained dual problems and an explicit solution with soft and hard bounds can be found. ¿ American Geophysical Union 1989