Solving Constrained Nonlinear Least Squares Problems Using an Exact Penalty Structured Projected Scheme with a Hybrid Trust Region-Line Search Procedure

We consider solving constrained nonlinear least squares problems using an exact penalty approach. The approach makes use of the adaptive structured scheme due to Mahdavi-Amiri and Bartels, and of the ideas of Nocedal and Overton for handling quasi-Newton updates of projected Hessians. We also present a new penalty parameter updating strategy and a hybridization of a line search strategy and a trust region method, accounting for the least squares objective function and the inherent structure of the approximate projected Hessians. These new features and the addition of a second order correction step in the global phase accelerate the global iterations in reaching the asymptotic region. The testing results of our programs on a variety of test problems show an obvious asymptotic two-step superlinear rate of convergence. The comparative results obtained by our approach, as compared to the ones obtained by Mahdavi-Amiri and Bartels, and three methods in KNITRO, show the effectiveness of our adaptive penalty updating scheme, the hybrid strategy, and special structured approximation of projected least squares Hessians