This paper deals with the optimal midcourse guidance maximizing the final velocity for the air-to-air missiles using convex optimization. To this end, a highly nonlinear optimal control problem is first converted into the second-order cone programming (SOCP) using successive linearization and convexification techniques. In the proposed approach, the angle-of-attack rate is chosen as a pseudo control input in order to obtain the control affine system. Additionally, linear regularization terms on the performance index are introduced to relieve the sub-problem infeasibility issue in the successive linearization procedure. Finally, numerical simulations are performed to validate the proposed approach.