This study proposes a new trust-region based sequential linear programming algorithm to solve the AC optimal power flow (OPF) problem. The OPF problem is solved by linearizing the cost function, power balance and engineering constraints of the system, followed by a trust-region to control the validity of the linear model. To alleviate the problems associated with the infeasibilities of a linear approximation, a feasibility restoration phase is introduced. This phase uses the original nonlinear constraints to quickly locate a feasible point when the linear approximation is infeasible. The algorithm follows convergence criteria to satisfy the first order optimality conditions for the original OPF problem. Studies on standard IEEE systems and large-scale Polish systems show an acceptable quality of convergence to a set of best-known solutions and a substantial improvement in computational time, with linear scaling proportional to the network size.