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.
Cite this paper
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.
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
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
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
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
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