General Trust Region Subproblem Research Articles

An affine scaling interior trust-region method combining with line search filter technique for optimization subject to bounds on variables

This paper proposes and analyzes an affine scaling trust-region method with line search filter technique for solving nonlinear optimization problems subject to bounds on variables. At the current iteration, the trial step is generated by the general trust-region subproblem which is defined by minimizing a quadratic function subject only to an affine scaling ellipsoidal constraint. Both trust-region strategy and line search filter technique will switch to trail backtracking step which is strictly feasible. Meanwhile, the proposed method does not depend on any external restoration procedure used in line search filter technique. A new backtracking relevance condition is given which is weaker than the switching condition to obtain the global convergence of the algorithm. The global convergence and fast local convergence rate of this algorithm are established under reasonable assumptions. Preliminary numerical results are reported indicating the practical viability and show the effectiveness of the proposed algorithm.

TW-TOA based positioning in the presence of clock imperfections

This manuscript studies the positioning problem based on two-way time-of-arrival (TW-TOA) measurements in semi-asynchronous wireless sensor networks in which the clock of a target node is unsynchronized with the reference time. Since the optimal estimator for this problem involves difficult nonconvex optimization, two suboptimal estimators are proposed based on the squared-range least squares and the least absolute mean of residual errors. We formulate the former approach as an extended general trust region subproblem (EGTR) and propose a simple technique to solve it approximately. The latter approach is formulated as a difference of convex functions programming (DCP), which can be solved using a concave–convex procedure. Simulation results illustrate the high performance of the proposed techniques, especially for the DCP approach.

An Affine Scaling Interior Trust-Region Algorithm Combining Backtracking Line Search with Filter Technique for Nonlinear Constrained Optimization

In this article, an affine scaling interior trust-region algorithm which employs backtracking line search with filter technique is presented for solving nonlinear equality constrained programming with nonnegative constraints on variables. At current iteration, the general full affine scaling trust-region subproblem is decomposed into a pair of trust-region subproblems in vertical and horizontal subspaces, respectively. The trial step is given by the solutions of the pair of trust-region subproblems. Then, the step size is decided by backtracking line search together with filter technique. This is different from traditional trust-region methods and has the advantage of decreasing the number of times that a trust-region subproblem must be resolved in order to determine a new iteration point. Meanwhile, using filter technique instead of merit function to determine a new iteration point can avoid the difficult decisions regarding the choice of penalty parameters. Under some reasonable assumptions, the new method possesses the property of global convergence to the first-order critical point. Preliminary numerical results show the effectiveness of the proposed algorithm.

RSS-Based Sensor Localization in the Presence of Unknown Channel Parameters

This correspondence studies the received signal strength-based localization problem when the transmit power or path-loss exponent is unknown. The corresponding maximum-likelihood estimator (MLE) poses a difficult nonconvex optimization problem. To avoid the difficulty in solving the MLE, we use suitable approximations and formulate the localization problem as a general trust region subproblem, which can be solved exactly under mild conditions. Simulation results show a promising performance for the proposed methods, which also have reasonable complexities compared to existing approaches.

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.

Projected Hessian algorithm with backtracking interior point technique for linear constrained optimization

In this paper, we propose a new trust-region-projected Hessian algorithm with nonmonotonic backtracking interior point technique for linear constrained optimization. By performing the QR decomposition of an affine scaling equality constraint matrix, the conducted subproblem in the algorithm is changed into the general trust-region subproblem defined by minimizing a quadratic function subject only to an ellipsoidal constraint. By using both the trust-region strategy and the line-search technique, each iteration switches to a backtracking interior point step generated by the trustregion subproblem. The global convergence and fast local convergence rates for the proposed algorithm are established under some reasonable assumptions. A nonmonotonic criterion is used to speed up the convergence in some ill-conditioned cases.

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.

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.

Interior point projected reduced Hessian method with trust region strategy for nonlinear constrained optimization

A interior point scaling projected reduced Hessian method with combination of nonmonotonic backtracking technique and trust region strategy for nonlinear equality constrained optimization with nonegative constraint on variables is proposed. In order to deal with large problems, a pair of trust region subproblems in horizontal and vertical subspaces is used to replace the general full trust region subproblem. The horizontal trust region subproblem in the algorithm is only a general trust region subproblem while the vertical trust region subproblem is defined by a parameter size of the vertical direction subject only to an ellipsoidal constraint. Both trust region strategy and line search technique at each iteration switch to obtaining a backtracking step generated by 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. A nonmonotonic criterion and the second order correction step are used to overcome Maratos effect and speed up the convergence progress in some ill-conditioned cases.

An affine scaling projective reduced Hessian algorithm for minimum optimization with nonlinear equality and linear inequality constraints

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.

An affine scaling interior point backtracking algorithm for nonlinear constrained optimisation

AbstractIn this paper we propose a new affine scaling interior trust region algorithm with a nonmonotonic backtracking technique for nonlinear equality constrained optimisation with nonnegative constraints on the variables. In order to deal with large problems, the general full trust region subproblem is decomposed into a pair of trust region subproblems in horizontal and vertical subspaces. The horizontal trust region subproblem in the algorithm is defined by minimising a quadratic function subject only to an ellipsoidal constraint in a null tangential subspace and the vertical trust region subproblem is defined by the least squares subproblem subject only to an ellipsoidal constraint. By adopting Fletcher's penalty function as the merit function, combining a trust region strategy and a nonmonotone line search, the mixing technique will switch to a backtracking step generated by the two trust region subproblems to obtain an acceptable step. The global convergence of the proposed algorithm is proved while maintaining a fast local superlinear convergence rate, which is established under some reasonable conditions. A nonmonotonic criterion is used to speed up the convergence progress in some highly nonlinear cases.

An interior affine scaling projective algorithm for nonlinear equality and linear inequality constrained optimization

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.