We propose a nonsmooth optimization method for $H_\infty$ output feedback control problem of linear time-invariant systems based on bundle technique. We formulate this problem as a nonconvex and nonsmooth semi-infinite constrained optimization problem by quantifying both internal stability of close-loop system and measurement of system performance, where $H_\infty$ norm of closed-loop transfer function and a stabilization channel is used. Our method uses progress function and bundle technique to solve the resulting problem which has a composite structure. The convergence to a critical point from a feasible initial point is proved and some benchmarks are tested to demonstrate the effectiveness of this method. We also extend this method to a general framework to solve constrained composite optimization problems.