This paper proposes a new trajectory optimization method for Unmanned Aerial Vehicle (UAV). The proposed L1-Penalized Sequential Convex Programming (LPSCP) method improves the initial infeasibility of the standard sequential convex programming. LPSCP method converges within 0.2 seconds which is more than one-tenth of the pseudospectral (PS) method. Therefore, LPSCP has the potential to enable UAV's real-time autonomous air mission if implemented on-board. The UAV trajectory optimization problem is defined at the beginning of the paper and a convexification process is performed when there are several no-fly zones along the trajectory. Then the LPSCP method iteratively solves locally approximated subproblems in conic forms. Simulation results illustrate the proposed method satisfies the required constraints and has a computation time advantage over the conventional PS method. Copyright (C) 2019. The Authors. Published by Elsevier Ltd. All rights reserved.