In collaboration with Payame Noor University and the Iranian Society of Instrumentation and Control Engineers

Document Type : Research Article


Department of Mathematics, Jahrom University, Jahrom, P.O. Box 74135-111, Iran


While many real-world optimization problems typically involve multiple constraints, unconstrained problems hold practical and fundamental significance. They can arise directly in specific applications or as transformed versions of constrained optimization problems.‎ ‎Newton's method‎, ‎a notable numerical technique within the category of line search algorithms, is widely used for function optimization‎. The search direction and step length play crucial roles in this algorithm. ‎This paper introduces an algorithm aimed at enhancing the step length within the Broyden quasi-Newton process‎. ‎Additionally‎, ‎numerical examples are provided to compare the effectiveness of this new method with another approach‎.


Main Subjects

[1] Andrei, N. (2008). “An unconstrained optimization test functions collection”, Advanced Modeling and Optimization, 10(1), 147-161.
[2] Broyden, C. (1965). “A class of methods for solving nonlinear simultaneous equations”, Mathematics of Computation, 19, 577-593.
[3] Dehghan, T., Niri, M., Hosseini, M., Heydari, M. (2019). “An efficient improvement of the Newton’s method for solving non-convex optimization problems”, Computational Methods for Differential Equations, 7(1), 69-85.
[4] Dennis, J.E., Schnabel, R.B. (1993). “Numerical methods for unconstrained optimization and non-linear equations”, SIAM, Philadelphia.
[5] Fang, X.W., Ni, Q., Zeng, M.L. (2018). “A modified quasi-Newton’s method for nonlinear equations”, Journal of Computational and Applied Mathematics, 328, 44-58.
[6] Li, D.H., Fukushima, M., Qi, L., Yamashita, N. (2004). “Regularized Newton methods for convex minimization problems with singular solutions”, Computational Optimization and Applications, 28, 131-147.
[7] Narcowich, F.J. (2023). “The Rank of a Matrix”,
[8] Polyak, R.A. (2009). “Regularized Newton’s method for unconstrained convex optimization”, Mathematical Programming, 120, 125-145.
[9] Sa, H., Chen, G.Q., Sui, Y.K., Wu, C.Y. (2016). “A new newton-like method for solving nonlinear equations”, Springer Plus., 5, 12-69.
[10] Shen, Ch., Xiongda, Ch., Liang, Y. (2012). “A regularized Newton’s method for degenerate unconstrained optimization problems”, Optimization Letters, 6, 1913-1933.
[11] Song, W.u., Haijun, W. (2020). “A modified Newton-like method for nonlinear equations”, Computational and Applied Mathematics, 39, 238.
[12] Sui, Y., Sa, H., Chen, G. (2014). “An improvement for the rational approximation RALND at accumulated two-point information”, Mathematica Numerica Sinica, 36, 51-64.
[13] Ueda, K., Yamashita, N. (2010). “Convergence properties of the regularized Newton’s method for the unconstrained non-convex optimization”, Applied Mathematics and Optimization, 62, 27-46.
[14] Yuan, G., Wang, Zh., Li, P. (2022). “Global convergence of a modified Broyden family method for non-convex functions”, Journal of Industrial and Management Optimization, 18(6), 4393-4407.
[15] Zhou, G., Toh, K.C. (2005). “Superlinear convergence of a Newton-type algorithm for monotone equations”, Journal of Optimization Theory and Applications, 125, 205-221.