Worst case complexity of direct search under convexity

In this paper we prove that the broad class of direct-search methods of directional type, based on imposing sufficient decrease to accept new iterates, exhibits the same global rate or worst case complexity bound of the gradient method for the unconstrained minimization of a convex and smooth function. More precisely, it will be shown that the number of iterations needed to reduce the norm of the gradient of the objective function below a certain threshold is at most proportional to the inverse of the threshold. Our result is slightly less general than Nesterov's for the gradient method, in the sense that we require more than just convexity of the objective function and boundedness of the initial iterate to the solution set. Our additional condition can, however, be satisfied in several scenarios, such as strong or uniform convexity, boundedness of the initial level set, or boundedness of the distance from the initial contour set to the solution set. It is a mild price to pay for deriving such a global rate for zero-order methods.


Preprint 13-10, Dept. Mathematics, Univ. Coimbra.



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