The phase-field approach to brittle fracture has emerged as a powerful tool to model crack initiation and propagation in complex geometries and loading scenarios. It represents the transition between intact and fully cracked material states by means of a phase-field variable that smoothly varies between 0 and 1. The model is based on the minimization of a total energy functional, depending on the displacement and on the phase field, under the constraint of damage irreversibility. The necessary conditions of this constrained minimization lead to a non-linear coupled system of governing equations. Due to the non-convexity of the energy functional, the most widely used solution procedure entails solving the two sets of equations governing equilibrium and damage evolution in a staggered fashion until convergence. This approach, termed alternate minimization or staggered scheme, is provably convergent and is generally considered very robust due to the separate convexity of the energy functional with respect to the displacement and the phase field. However, the decomposition of the elastic strain energy density into damage-driving and damage-resisting components, used to obtain an asymmetric behavior in tension and compression, and the irreversibility constraint introduce strong non-linearities which pose numerical challenges. First, the number of iterations needed to reach convergence may be very high, leading to a computational cost that limits the applicability of the approach to large systems. Secondly, when the iterative Newton-Raphson scheme is used to solve each of the two discretized governing equations, convergence issues may arise, especially in cases with multi-axial stress states with predominant compression and/or unstable crack propagation. A possible strategy to overcome such difficulties is the adoption of a line-search approach. Against this background, in this work we investigate and compare the robustness of the alternate minimization scheme enhanced by different line-search algorithms. We adopt a set of particularly challenging benchmark problems and compare different strain energy density decompositions. We highlight cases where convergence issues can be observed and demonstrate how line-search methods are able to overcome them, thus significantly improving robustness and computational efficiency of the solution algorithm. We also discuss the integration of these techniques into scalable computational frameworks.