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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, a derivative-free affine scaling inexact Levenberg–Marquardt method with interior backtracking line search technique is considered for solving linear inequality constrained nonlinear systems. The proposed algorithm is designed to take advantage of the problem structure by building polynomial interpolation models for each function of nonlinear systems subject to the linear inequality constraints on variables. Each iterate switches to backtracking step generated by affine scaling inexact Levenberg–Marquardt method and satisfies strict interior point feasibility by line search backtracking technique. Under local error bounded assumption, the method is superlinear and quadratic convergent on F ( x ). The results of numerical experiments are reported to show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents an affine scaling optimal path approach in association with nonmonotonic interior backtracking line search technique for nonlinear optimization subject to linear constraints. We shall employ the eigensystem decomposition and affine scaling mapping to form affine scaling optimal curvilinear path very easily. By using interior backtracking line search technique, each iterate switches to trial step of strict interior feasibility. The nonmonotone criterion is used to speed up the convergence progress in the contours of objective function with large curvature. Theoretical analysis is given which prove that the proposed algorithm is globally convergent and has a local superlinear convergence rate under some reasonable conditions. The results of numerical experiments are reported to show the effectiveness of the proposed algorithm. [Copyright &y& Elsevier]

Abstract: In this paper we propose an affine scaling interior algorithm via Lanczos path for solving nonlinear equality systems subject to bounds on variables. Employing the affine scaling Lanczos path search strategy, we obtain an iterative direction by solving the linearize model. By using the line search backtracking technique, we will find an acceptable trial step length along this direction which is strictly feasible and makes the objective function nonmonotonically decreasing. The global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions. Furthermore, the numerical results of the proposed algorithm indicate to be effective. [Copyright &y& Elsevier]

Abstract: This paper presents an affine scaling reduced preconditional conjugate gradient path approach in association with nonmonotonic interior backtracking line search technique for the linear constrained optimization. Employing the affine scaling preconditional conjugate gradient to form the curvilinear path and using interior backtracking line search technique, each iterate switches to trial step of strict interior feasibility. The nonmonotone criterion is used to speed up the convergence progress in the contours of objective function with large curvature. Theoretical analysis are given which prove that the proposed algorithm is globally convergent and has a local superlinear convergence rate under some reasonable conditions. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a new interior point algorithm for linear programming by combining the affine scaling algorithm and the primal path-following algorithm. We define a series of subproblems whose feasible region contracts toward the set of optimal solutions. In each iteration, we take a step in the steepest descent direction in a transformed space to reduce the objective value. Our preliminary computational results are very favorable. [Copyright &y& Elsevier]

Abstract: In this paper we propose a nonmonotonic interior point backtracking strategy to modify the reduced projective affine scaling trust region algorithm for solving minimum optimization subject to both nonlinear equality and linear inequality constraints. The general full trust region subproblem for solving the minimum optimization is decomposed to a pair of trust region subproblems in horizontal and vertical subspaces of linearize equality constraints and extended affine scaling equality constraints by QR decomposition of an affine scaling matrix and an orthonormal basis on the null subspace. The horizontal subproblem in the proposed algorithm is defined by minimizing a quadratic projective reduced Hessian function subject only to an ellipsoidal trust region constraint, while the vertical subproblem is also defined by the least squares subproblem subject only to an ellipsoidal trust region constraint. Combining trust region strategy with line search technique will switch to strictly feasible interior point step generated by a component direction of the two trust region subproblems. By adopting the l 1 penalty function as the merit function, the global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions. The second-order correction step and a nonmonotonic criterion are used to overcome Maratos effect and speed up the convergence progress in some ill-conditioned cases, respectively. [Copyright &y& Elsevier]

In this paper, we propose a new nonmonotonic interior point backtracking strategy to modify the reduced projective affine scaling trust region algorithm for solving optimization subject to nonlinear equality and linear inequality constraints. The general full trust region subproblem for solving the nonlinear equality and linear inequality constrained optimization is decomposed to a pair of trust region subproblems in horizontal and vertical subspaces of linearize equality constraints and extended affine scaling equality constraints. The horizontal subproblem in the proposed algorithm is defined by minimizing a quadratic projective reduced Hessian function subject only to an ellipsoidal trust region constraint in a null subspace of the tangential space, while the vertical subproblem is also defined by the least squares subproblem subject only to an ellipsoidal trust region constraint. By introducing the Fletcher''s penalty function as the merit function, trust region strategy with interior point backtracking technique will switch to strictly feasible interior point step generated by a component direction of the two trust region subproblems. The global convergence of the proposed algorithm while maintaining fast local convergence rate of the proposed algorithm are established under some reasonable conditions. A nonmonotonic criterion should bring about speeding up the convergence progress in some high nonlinear function conditioned cases. [Copyright &y& Elsevier]

In this paper we propose a new interior affine scaling region algorithm with nonmonotonic interior point backtracking technique for nonlinear optimization subject to linear equality and inequality constraints. The trust region subproblem in the proposed algorithm is defined by minimizing a quadratic function subject only to an affine scaling ellipsoidal constraint in a null subspace of the extended equality constraints. Using both trust region strategy and line search technique, the affine scaling trust region subproblem at each iteration generates backtracking interior step to obtain a new accepted step. The global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions. A nonmonotonic criterion should bring about speeding up the convergence progress in some ill-conditioned cases. [Copyright &y& Elsevier]