A Derivative-free Algorithm For Nonlinear Least-squares Problems With Bound Constraints

In the fields of physics, biology, economics, architecture, circuit design and aerospace, there are a lot of nonlinear least-squares problems like a “black box”.We can hardly get specific function expressions and can only get the results through experiments or simulations. It is almost impossible for us to get their derivative values. To solve these particular cases, we design an algorithm on the basis of Powell’s derivative-free method and take fully advantage of the structure of least-squares problems. Function approximation approach and trust region method are used. We generate feasible iteration step by truncated conjugate gradient method and projection operator. In addition, we design a therapy procedure to adjust model to reduce errors and make algorithm more efficient.What’s more, when rounding error is arbitrarily large, which can cause serious bias in approximation model, we reselect the interpolation points and reconstruct the model by a procedure named remedy. Compared with quasi-Newton method,our algorithm is still robust with the presence of noise. Besides, it is especially suitable to solve problems whose function calculation is expensive. In numerical experiments, we compare our algorithm with Matlab’s in the case of having noise or not, respectively. In addition, we test the efficiency of the algorithm by CUTEst, which is a real “black box”.