In this paper, we suggest and analyze some new derivative free iterative methods for solving nonlinear equation using a trust-region method. We also, give several examples to illustrate the efficiency of these methods. Comparison with other similar method is also given. This tech-nique can be used to suggest a wide class of new iterative methods for solving optimization problem. For, solving linearly unconstrained optimi-zation problems without derivatives, a derivative-free Funnel method for unconstrained non-linear optimization is proposed. The study presents new interpolation-based techniques. The main work of this paper depends on some matrix computation techniques. A linear system is solved to obtain the required quadratic model at each iteration. Interpolation points are based on polynomial which is then minimized in a trust-region.
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Mu’lla, M. A. M. (2019). An Algorithm for the Derivative-Free Unconstrained Optimization Based on a Moving Random Cone Data Set. Open Access Library Journal, 6, e5652. doi: http://dx.doi.org/10.4236/oalib.1105652.
Nocedal, J. and Wright, S.J. (2000) Numerical Optimization Mathematics Subject Classification. 2nd Edition, Library of Congress Control Number: 2006923897. https://doi.org/10.1007/b98874
Wilkinson, J.H. (1960) Householder’s Method for the Solution of the Algebraic Eigenproblem. The Computer Journal, 3, 23-27. https://doi.org/10.1093/comjnl/3.1.23
Powell, M.J.D. (1970) A Hybrid Method for Nonlinear Equations. In: Robinowitz, P., Ed., Numerical Methods for Nonlinear Algebraic Equations, Gordon and Breach Science, London, 87-144.
Conn, A.R., Gould, N.I.M. and Toint, P.L. (2000) Trust-Region Methods. MPS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics, Philadelphia, PA. https://doi.org/10.1137/1.9780898719857
Dennis, J.E. and Schnabel, A.B. (1989) A View of Unconstrained Optimization. In: Handbooks in Operations Research and Management, Elsevier Science Publishers, Amsterdam, The Netherlands, 1-72. https://doi.org/10.1016/S0927-0507(89)01002-9
Golub, G.H. and von Matt, U. (1991) Quadratically Constrained Least Squares and Quadratic Problems. Numerische Mathematik, 59, 561-580. https://doi.org/10.1007/BF01385796
Gratton, S., Toint, P.L. and Troltzsch, A. (2011) An Active-Set Trust-Region Method for Derivative-Free Nonlinear Bound-Constrained Optimization. Optimization Methods and Software, 26, 873-894.
Powell, M.J.D. (1970) A New Algorithm for Unconstrained Optimization. In: Rosen, J.B., Mangasarian, O.L. and Ritter, K., Eds., Nonlinear Programming, Academic Press, New York, 31-65. https://doi.org/10.1016/B978-0-12-597050-1.50006-3
Moré, J.J. and Sorenson, D.C. (1983) Computing a Trust Region Step. SIAM Journal on Scientific and Statistical Computing, 4, 553-572. https://doi.org/10.1137/0904038
Ciarlet, P.G. and Raviart, P.A. (1972) General Lagrange and Hermite Interpolation in 〖IR〗^n with Applications to Finite Element Methods. Archive for Rational Mechanics and Analysis, 46, 177-199. https://doi.org/10.1007/BF00252458
Erd?s, P. (1961) Problems and Results on the Theory of Interpolation. II. Acta Mathematica Academiae Scientiarum Hungarica, 12, 235-244. https://doi.org/10.1007/BF02066686
Byrd, R.H., Schnabel, R.B. and Schultz, A.A. (1988) Approximate Solution of the Trust Regions Problem by Minimization over Two-Dimensional Subspaces. Mathematical Programming, 40, 247-263. https://doi.org/10.1007/BF01580735
Conn, A.R., Scheinberg, K. and Vicente, L.N. (2008) Introduction to Derivative-Free Optimization. MPS-SIAM Series on Optimization. SIAM, Philadelphia, PA. https://doi.org/10.1137/1.9780898718768
Dennis, J.E. and Mei, H.H.W. (1979) Two New Unconstrained Optimization Algorithms Which Use Function and Gradient Values. Journal of Optimization Theory and Applications, 28, 453-482. https://doi.org/10.1007/BF00932218
Omojokun, E.O. (1989) Trust Region Algorithms for Optimization with Nonlinear Equality and Inequality Constraints. Ph.D. Thesis, University of Colorado, Boulder, CO.