Incorporating stress constraints in topology optimization is a challenging task due to the large number of constraints in the formulation. One effective strategy to address this challenge is the augmented Lagrangian (AL) method, which transforms the original stress-constrained problem into a sequence of subproblems with only bound constraints. The effectiveness of the AL method heavily depends on the optimization method used to solve these subproblems. This work performs a comparative study of six optimization solvers: The method of moving asymptotes (MMA), the steepest descent method with move limits (SDM), the spectral projected gradient (SPG), and the limited-memory BFGS with bound constraints (L-BFGS-B), along with two proposed adaptations, steepest descent method with move limits—Barzilai–Borwein (SDMBB) and spectral projected gradient with move limits (SPGM). These methods are evaluated in the context of the volume minimization problem with local stress constraints. The solutions are compared in terms of performance, defined as the final volume fraction, and efficiency, measured by the number of state and adjoint analyses required. A mesh dependence study is conducted to assess the robustness of each method across different mesh sizes, including high-resolution cases with approximately 1.8 million elements. SDMBB exhibits the highest efficiency, while SPGM achieves the best performance, followed by SDM. The MMA, SPG, and L-BFGS-B show limitations in high-resolution problems or fail to meet specific stopping criteria. The results demonstrate that the choice of the optimization solver significantly affects the efficiency of the AL method, as well as the performance and mesh dependence of the solutions. Furthermore, this study identifies the most promising methods for solving large-scale stress-constrained problems.