This paper studies predefined-time convergent distributed fault-tolerant optimization algorithms for the resource allocation problem of multi-agent systems with actuator faults. To address the impact of heterogeneous control gains caused by partial loss of effectiveness faults and unknown disturbances caused by biased faults on system dynamics, this paper first proposes a fault-tolerant distributed weighted gradient algorithm based on output feedback and variable structure control methods. Using tools such as the predefined-time Lyapunov stability theory and convex optimization theory, the predefined-time optimal convergence stability of the algorithm is analyzed. Then, to avoid input chattering caused by discontinuous variable structure control, a fault-tolerant distributed weighted gradient algorithm with continuous control inputs is designed, and its practical predefined-time stability is analyzed. Finally, simulation results demonstrate the effectiveness of the proposed fault-tolerant optimization algorithms.