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We introduced an algorithm for unconstrained optimization based on the transformation of the Newton method with the line search into a gradient descent method. Main idea used in the algorithm construction is approximation of the Hessian by an appropriate diagonal matrix. The steplength calculation algorithm is based on the Taylor's development in two successive iterative points and the backtracking line search procedure. The linear convergence of the algorithm is proved for uniformly convex functions and strictly convex quadratic functions satisfying specified conditions.

An iterative method for solving a minimization problem of unconstrained optimization is presented. This multistep curve search method uses the specific form of iteration with two direction parameters, the approximation of Hessian by appropriately constructed diagonal matrix, and the inexact line search procedure. It is proved that constructed numerical process is well defined under some assumptions. Considering certain conditions, the method is linearly convergent for uniformly convex and strictly convex quadratic functions. Numerical results arising from defined algorithms are also presented and analyzed.

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