Interior Backtracking Line Search Research Articles

An inexact derivative-free Levenberg–Marquardt method for linear inequality constrained nonlinear systems under local error bound conditions

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.

An affine-scaling derivative-free trust-region method for solving nonlinear systems subject to linear inequality constraints

In this paper, an affine-scaling derivative-free trust-region method with interior backtracking line search technique is considered for solving nonlinear systems subject to linear inequality constraints. The proposed algorithm is designed to take advantage of the problem structured by building polynomial interpolation models for each function in the nonlinear system function F. The proposed approach is developed by forming a quadratic model with an appropriate quadratic function and scaling matrix: there is no need to handle the constraints explicitly. By using both trust-region strategy and interior backing line search technique, each iteration switches to backtracking step generated by the trust-region subproblem and satisfies strict interior point feasibility by line search backtracking technique. Under reasonable conditions, the global convergence and fast local convergence rate of the proposed algorithm are established. The results of numerical experiments are reported to show the effectiveness of the proposed algorithms.

A trust-region algorithm combining line search filter method with Lagrange merit function for nonlinear constrained optimization

This paper presents a trust-region algorithm which employs line search filter technique with Lagrange merit function for solving nonlinear equality constrained programming. At the current iteration, the general full trust-region subproblem is decomposed into a pair of trust-region subproblems in normal and tangential subspaces. The trial step is given by solving these two trust-region subproblems. Then, different from traditional trust-region method in which the next iterate is determined by the ratio of the actual reduction to the predicted reduction, the step size is decided by interior backtracking line search together with filter method. Consequently, the expensive computation raised by resolving trust-region subproblem many times to determine a new iteration point in traditional trust-region method can be reduced. And the difficult decisions in regard to the choice of penalty parameters in the merit functions can be avoided by using filter technique. The new method retains the global convergence to the first-order critical points under some reasonable assumptions without using a merit function. Meanwhile, by using Lagrange function in the filter, the method can overcome Maratos effect without using second order correction step and hence the superlinear local convergence is achieved. The preliminary numerical results are reported to show effectiveness of the proposed algorithm.

A trust-region algorithm combining line search filter technique for nonlinear constrained optimization

In this paper, we propose a trust-region algorithm in association with line search filter technique for solving nonlinear equality constrained programming. At current iteration, a trial step is formed as the sum of a normal step and a tangential step which is generated by trust-region subproblem and the step size is decided by interior backtracking line search together with filter methods. Then, the next iteration is determined. This is different from general trust-region methods in which the next iteration is determined by the ratio of the actual reduction to the predicted reduction. The global convergence analysis for this algorithm is presented under some reasonable assumptions and the preliminary numerical results are reported.

Inexact Newton method via Lanczos decomposed technique for solving box-constrained nonlinear systems

This paper proposes an inexact Newton method via the Lanczos decomposed technique for solving the box-constrained nonlinear systems. An iterative direction is obtained by solving an affine scaling quadratic model with the Lanczos decomposed technique. By using the interior backtracking line search technique, an acceptable trial step length is found along this direction. The global convergence and the fast local convergence rate of the proposed algorithm are established under some reasonable conditions. Furthermore, the results of the numerical experiments show the effectiveness of the proposed algorithm.

Superlinearly convergent affine scaling interior trust-region method for linear constrained LC1 minimization

We extend the classical affine scaling interior trust region algorithm for the linear constrained smooth minimization problem to the nonsmooth case where the gradient of objective function is only locally Lipschitzian. We propose and analyze a new affine scaling trust-region method in association with nonmonotonic interior backtracking line search technique for solving the linear constrained LC1 optimization where the second-order derivative of the objective function is explicitly required to be locally Lipschitzian. The general trust region subproblem in the proposed algorithm is defined by minimizing an augmented affine scaling quadratic model which requires both first and second order information of the objective function subject only to an affine scaling ellipsoidal constraint in a null subspace of the augmented equality constraints. The global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions where twice smoothness of the objective function is not required. Applications of the algorithm to some nonsmooth optimization problems are discussed.

An affine scaling optimal path method with interior backtracking curvilinear technique for linear constrained optimization

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.

Nonmonotonic Reduced Projected Hessian Method Via an Affine Scaling Interior Modified Gradient Path for Bounded-Constrained Optimization

The authors propose an affine scaling modified gradient path method in association with reduced projective Hessian and nonmonotonic interior backtracking line search techniques for solving the linear equality constrained optimization subject to bounds on variables. By employing the QR decomposition of the constraint matrix and the eigensystem decomposition of reduced projective Hessian matrix in the subproblem, the authors form affine scaling modified gradient curvilinear path very easily. By using interior backtracking line search technique, each iterate switches to trial step of strict interior feasibility. The global convergence and fast local superlinear/quadratical convergence rates 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. The results of numerical experiments are reported to show the effectiveness of the proposed algorithm.

Affine scaling interior Levenberg–Marquardt method for bound-constrained semismooth equations under local error bound conditions

We develop and analyze a new affine scaling Levenberg–Marquardt method with nonmonotonic interior backtracking line search technique for solving bound-constrained semismooth equations under local error bound conditions. The affine scaling Levenberg–Marquardt equation is based on a minimization of the squared Euclidean norm of linear model adding a quadratic affine scaling matrix to find a solution that belongs to the bounded constraints on variable. The global convergence results are developed in a very general setting of computing trial directions by a semismooth Levenberg–Marquardt method where a backtracking line search technique projects trial steps onto the feasible interior set. We establish that close to the solution set the affine scaling interior Levenberg–Marquardt algorithm is shown to converge locally Q-superlinearly depending on the quality of the semismooth and Levenberg–Marquardt parameter under an error bound assumption that is much weaker than the standard nonsingularity condition, that is, BD-regular condition under nonsmooth case. A nonmonotonic criterion should bring about speed up the convergence progress in the contours of objective function with large curvature.

An affine scaling reduced preconditional conjugate gradient path method for linear constrained optimization

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.

An affine scaling interior trust-region method for LC1 minimization subject to bounds on variables

In this literature, we extend the classical affine scaling interior trust-region algorithm for smooth bounded-constrained nonlinear programming to the nonsmooth case where the objective function is only locally Lipschitzian. We propose and analyze a new affine scaling trust-region method in association with nonmonotonic interior backtracking line search technique for solving the linear equality constrained LC^1 optimization subject to bounds on variables. At each iteration, the trust-region subproblem is defined by minimizing a quadratic function which is required both first- and second-order information of the objective function and a quadratic affine scaling matrix model subject only to an ellipsoidal constraint in a null space of the affine scaling linear equality constraints. The second-order derivative of the objective function is explicitly required to be Lipschitzian. Each iterate switches to a very general setting of computing trial backtracking steps generated by the general trust-region subproblem and then the line search technique projects the trial backtracking steps onto the strictly feasible set of the bounded constraints on variables. The global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions where twice smoothness of the objective function is not required. A nonmonotonic criterion should bring about speeding up the convergence progress in some ill-conditioned cases. Applications of the algorithm to various nonsmooth optimization problems are discussed.

Affine scaling inexact generalized Newton algorithm with interior backtracking technique for solving bound-constrained semismooth equations

We develop and analyze an affine scaling inexact generalized Newton algorithm in association with nonmonotone interior backtracking line technique for solving systems of semismooth equations subject to bounds on variables. By combining inexact affine scaling generalized Newton with interior backtracking line search technique, each iterate switches to inexact generalized Newton backtracking step to strict interior point feasibility. The global convergence results are developed in a very general setting of computing trial steps by the affine scaling generalized Newton-like method that is augmented by an interior backtracking line search technique projection onto the feasible set. Under some reasonable conditions we establish that close to a regular solution the inexact generalized Newton method is shown to converge locally p-order q-superlinearly. We characterize the order of local convergence based on convergence behavior of the quality of the approximate subdifferentials and indicate how to choose an inexact forcing sequence which preserves the rapid convergence of the proposed algorithm. A nonmonotonic criterion should bring about speeding up the convergence progress in some ill-conditioned cases.

An affine scaling trust-region algorithm with interior backtracking technique for solving bound-constrained nonlinear systems

In this paper, we propose a new affine scaling trust-region algorithm in association with nonmonotonic interior backtracking line search technique for solving nonlinear equality systems subject to bounds on variables. The trust-region subproblem is defined by minimizing a squared Euclidean norm of linear model adding the augmented quadratic affine scaling term subject only to an ellipsoidal constraint. By using both trust-region strategy and interior backtracking line search technique, each iterate switches to backtracking step generated by the general trust-region subproblem and satisfies strict interior point feasibility by line search backtracking technique. 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. The results of numerical experiments are reported to show the effectiveness of the proposed algorithm.