The mesh adaptive direct search algorithm (MADS) is an iterative method for constrained blackbox optimization problems. One of the optional MADS features is a versatile search step in which quadratic models are built leading to a series of quadratically constrained quadratic subproblems. This work explores different algorithms that exploit the structure of the quadratic models: the first one applies an l1-exact penalty function, the second uses an augmented Lagrangian and the third one combines the former two, resulting in a new algorithm. It is notable that this latter approach is uniquely suitable for quadratically constrained quadratic problems. These methods are implemented within the NOMAD software package and their impact are assessed through computational experiments on 65 analytical test problems and 4 simulation-based engineering applications.