A Second-Order Finite-Difference Method for Derivative-Free Optimization

Algorithm 1

Finite-difference algorithm.

	Input: Given initial iteration , the constants , , , and . Set .
	Main Step:
(1)	Choose , calculate the approximate gradient using (3) or (2). Choose , calculate the approximate Hessian matrix using (5) or (4) and calculate , where is the minimum eigenvalue of symmetric matrix .
(2)	If , then let . If , then stop, is the second-order stationary point, else go to 1.
(3)	If , then calculate the following subproblem,
	where and obtain the search direction .
(4)	Calculate
(5)	If then , else let .
(6)	Update the trust-region radius as follows:
	Let , go to 1.