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In this paper, we introduce an affine scaling cubic regularization algorithm for solving optimization problem without available derivatives subject to bound constraints employing a polynomial interpolation approach to handle the unavailable derivatives of the original objective function. We first define an affine scaling cubic model of the approximate objective function which is obtained by the polynomial interpolation approach with an affine scaling method. At each iteration a candidate search direction is determined by solving the affine scaling cubic regularization subproblem and the new iteration is strictly feasible by way of an interior backtracking technique. The global convergence and local superlinear convergence of the proposed algorithm are established under some mild conditions. Preliminary numerical results are reported to show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we modify the trust region interior point algorithm proposed by Bonnans and Pola in (SIAM J. Optim. 7(3) (1997) 717) for linear constrained optimization. A mixed strategy using both trust region and line-search techniques is adopted which switches to back-tracking steps when a trial step produced by the trust region subproblem may be unacceptable. The global convergence and local convergence rate of the improved algorithm are established under some reasonable conditions. A nonmonotonic criterion is used to speed up the convergence progress in some ill-conditioned cases. The results of numerical experiments are reported to show the effectiveness of the proposed algorithm. [Copyright &y& Elsevier]