For the last decade, research using convex optimization has been attempted for various problems in the aerospace engineering field. Since the convex optimization is robust to initial conditions and ensures convergence within the polynomial time, there are several papers on trajectory optimization problems for powered descent guidance in Mars exploration field where the initial condition is uncertain. In the case of air-to-ground missiles, the approach using convex optimization is effective to generate the optimal trajectory quickly and robustly because of various initial conditions and complex constraints in engagement scenarios within the operating range of a platform. This paper deals with the method of optimizing the mid-course trajectory of air-to-ground missiles using convex programming approaches. The problem of minimizing total energy and maximizing final velocity of mid-course trajectory, which are frequently used in the mid-course guidance phase, is addressed with the maximum altitude constraint. However, since the missile trajectory optimization problem is generally described as a nonlinear optimization problem due to dynamic equations and non-convex constraints, an appropriate transformation is required to apply the convex optimization technique. First, in order to enhance the covergence capability, we applied the method of control dynamics augmentation or control variable augmentation, and then applied the partial linearization method to the nonlinear equation and discretized it based on the trapezoidal rule. In addition, to cope with the non-convex control constraint that occurs in the control variable augmentation problem, the lossless convexification is proved using the maximum principle of the optimal control theory. Finally, simulation results of the proposed convex programming approaches are compared with the results of nonlinear programming to confirm the effectiveness and robustness.