This paper investigates the use of Primal-Dual optimization algorithms on multidimensional signal processing problems. The data blocks interpreted in a tensor way can be modeled by means of multi-linear decomposition. Here we will focus on the Canonical Polyadic Decomposition (CPD), and we will present an application to fluorescence spectroscopy using this decomposition. In order to estimate the factors or latent variables involved in these decompositions, it is usual to use criteria optimization algorithms. A classical cost function consists of a measure of the modeling error (fidelity term) to which a regularization term can be added if necessary. Here, we consider one of the most efficient optimization methods, Primal-Dual Projected Gradient. The effectiveness and the robustness of the proposed approach are shown through numerical examples.