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Using Taylor's series, we propose a modified secant relation to get a more accurate approximation of the second curvature of the objective function. Then, using this relation and an approach introduced by Dai and Liao, we present a conjugate gradient algorithm to solve unconstrained optimization problems. The proposed method makes use of both gradient and function values, and utilizes information from the two most recent steps, while the usual secant relation uses only the latest step information. Under appropriate conditions, we show that the proposed method is globally convergent without needing convexity assumption on the objective function. Comparative results show computational efficiency of the proposed method in the sense of the Dolan-Moré performance profiles.

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Numerical solution of nonlinear least-squares problems is an important computational task in science and engineering. Effective algorithms have been developed for solving nonlinear least squares problems. The structured secant method is a class of efficient methods developed in recent years for optimization problems in which the Hessian of the objective function has some special structure. A primary and typical application of the structured secant method is to solve the nonlinear least squares problems. We present an exact penalty method for solving constrained nonlinear leastsquares problems, when the structured projected Hessian is approximated by a projected version of the structured BFGS formula and give its local two-step Q-superlinear convergence. For robustness, we employ a special nonsmooth line search strategy, taking account of the least squares objective. We discuss the comparative results of the testing of our programs and three nonlinear programming codes from KNITRO on some randomly generated test problems due to Bartels and Mahdavi-Amiri. Numerical results also confirm the practical relevance of our special considerations for the inherent structure of the least squares.