This article introduces a new Penalized Majorization-Minimization Subspace algorithm (P-MMS) for solving smooth, constrained optimization problems. In short, our approach consists of embedding a subspace algorithm in an inexact exterior penalty procedure. The subspace strategy, combined with a Majoration-Minimization step-size search, takes great advantage of the smoothness of the penalized cost function, while the penalty method allows to handle a wide range of constraints. The main drawback of exterior penalty approaches, namely ill-conditioning for large values of the penalty parameter, is overcome by using a trust-region like technique. The convergence of the resulting algorithm is analyzed. Numerical experiments carried out on two large-scale image recovery applications demonstrate that, compared with state-of-the-art algorithms, the proposed method performs well in terms of computational time.