Convergence and solution time are important considerations in process system optimization. Another nontrivial task is the definition of termination criteria. However, setting the convergence tolerance is difficult and bewildering for users. Observed behaviors of algorithms when solving many optimization problems include tardiness in deciding convergence or failure of the optimization, and incapability of giving approximate solutions as they fail to converge. Here, we propose convergence depth control (CDC) for process system optimization. It is designed to take advantage of the achievement estimation of the optimization process to discover the proper time to terminate the optimization algorithm. Criteria based on CDC prefer to provide an approximate solution with acceptable optimality. Achievability and rationality of the criteria have been analyzed. To demonstrate the effectiveness of this method, we apply the Reduced-Hessian Successive Quadratic Programming (RSQP) algorithm with convergence depth control and with traditional convergence criteria, respectively, to problems from the CUTE test set, the distillation sequence in ethylene production, and catalyst mixing problem in COPS collection. Numerical results of the comparison show significant advantages of convergence depth control.