Volume: 18 • Number: 1 • Pages: 195-213

This manuscript presents a derivative-free quadratic regularization method for unconstrained minimization of a smooth function with Lipschitz continuous gradient. At each iteration, trial points are computed by minimizing a quadratic regularization of a local model of the objective function. The models are based on forward finite-difference gradient approximations. By using a suitable acceptance condition for the trial points, the accuracy of the gradient approximations is dynamically adjusted as a function of the regularization parameter used to control the stepsizes. Worst-case evaluation complexity bounds are established for the new method. Specifically, for nonconvex problems, it is shown that the proposed method needs at most function evaluations to generate an -approximate stationary point, where n is the problem dimension. For convex problems, an evaluation complexity bound of is obtained, which is reduced to under strong convexity. Numerical results illustrating the performance of the proposed method are also reported