Distributed optimization control represents a classic problem in multi-agent systems. However, achieving rapid convergence under unbalanced graphs remains a highly challenging issue. By constructing two time-varying gain functions, this paper proposes a dual-stage distributed optimization control framework, namely distributed predefined-time estimation and distributed predefined-time optimization control. Compared with existing studies on distributed optimization control for heterogeneous linear multi-agent systems, the proposed algorithm simultaneously possesses the following advantages: 1) faster convergence, it can converge to the exact optimal solution within a predefined-time, where the convergence time is only related to a single time parameter, independent of the initial system state and control parameters, facilitating easy adjustment of the convergence time? 2) bounded control inputs, the algorithm does not require conventional fractional-power feedback, leading to a simpler and more concise structure; 3) applicability to more general unbalanced communication networks. Finally, the effectiveness and superiority of the algorithm are rigorously verified through Lyapunov-based analysis and numerical simulations.