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This paper investigates the control barrier function (CBF) based safety-critical control for continuous nonlinear control affine systems using the more efficient online algorithms through time-varying optimization. The idea lies in that when quadratic programming (QP) or other convex optimization algorithms needed in the CBF-based method is not computation affordable, the alternative suboptimal feasible solutions can be obtained more economically. By using the barrier-based interior point method, the constrained CBF-QP problems are converted into the unconstrained ones with suboptimal solutions tracked by two continuous descent-based algorithms. Considering the lag effect of tracking and exploiting the system information, the prediction method is added to the algorithms which thereby achieves a exponential convergence rate to the time-varying suboptimal solutions. The convergence and robustness of the designed methods as well as the safety criteria of the algorithms are analyzed theoretically. In the end, the effectiveness is illustrated by simulations on the anti-swing and obstacle avoidance tasks.