Recently, the so-called (model-free) data-driven computational mechanics (DDCM) has established an alternative paradigm allowing to completely bypass explicit constitutive models. Such method. The DDCM solver relies exclusively on a finite dataset of gradient-fluxes pairs and consists of two steps : i) a continuous constrained minimisation problem and ii) a nearest neighbour search (of finite nature). In this work, we explore alternative choices to rephrase Problem i) into an unconstrained format via Lagrange Multipliers, leading to different DDCM algorithms.