Primal-dual gradient methods are tractable approaches to solve quadratic programs, especially for large scale problems with sparse structures. The associated iterative mechanism allows these methods to converge to the optimal solution. However, the convergence may require a considerable number of iterations, and if one decides to terminate the iterative process earlier, the resulting solution may not fulfill the original constraints of the problem. For the first issue, we propose a proactive method based on the Karush–Kuhn–Tucker (KKT) conditions to check whether the active set updated during the iterations is optimal, which can terminate the iterations before its convergence. To address the second issue and faster terminate the iterations, we first introduce a degree of suboptimality for objective value, and then propose a suboptimal method to solve for suboptimal and feasible solutions. Mathematical developments prove both the feasibility and the guarantee of predefined suboptimality for the suboptimal method. Various random simulations illustrate the effectiveness of the proposed methods.