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This paper proposes an affine scaling interior trust-region method in association with nonmonotone line search filter technique for solving nonlinear optimization problems subject to linear inequality constraints. Based on a Newton step which is derived from the complementarity conditions of linear inequality constrained optimization, a trust-region subproblem subject only to an ellipsoidal constraint is defined by minimizing a quadratic model with an appropriate quadratic function and scaling matrix. The nonmonotone schemes combining with trust-region strategy and line search filter technique can bring about speeding up the convergence progress in the case of high nonlinear. A new backtracking relevance condition is given which assures global convergence without using the switching condition used in the traditional line search filter technique. The fast local convergence rate of the proposed algorithm is achieved which is not depending on any external restoration procedure. The preliminary numerical experiments are reported to show effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 propose a derivative-free restrictively preconditioned conjugate gradient path method without line search technique for solving linear equality constrained optimization. This method originates from the classical conjugate gradient method and its restrictively preconditioned variant. The preconditioned conjugate gradient path is obtained by solving extended system of the preconditioned equation with restrictively preconditioned conjugate gradient method which is a more robust and effective technique to solve large sparse linear system. The global convergence and local superlinear convergence rate of the proposed method are established under some reasonable conditions. Finally, the numerical results are reported to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose conjugate gradient path method for solving derivative-free unconstrained optimization. The iterative direction is obtained by constructing and solving quadratic interpolation model of the objective function with conjugate gradient methods. The global convergence and local superlinear convergence rate of the proposed algorithm are established under some reasonable conditions. Finally, the numerical results are reported to show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, we develop dwindling filter idea and trust region strategy for solving nonlinear optimization. Compared with traditional filter trust region algorithm, the new filter algorithm has more flexibility for the acceptance of the trial step and the feasibility restoration phase is not needed. Under mild conditions, the global convergence of the new algorithm is analyzed. Preliminary numerical results are reported. [Copyright &y& Elsevier]

Abstract: We present a new projected affine-scaling interior-point Newton’s method with a line search filter algorithm for nonlinear optimization problems with box constraints. The global convergence of the proposed algorithm is established under some reasonable conditions. Furthermore, the method is shown to be locally quadratically convergent under the strong second order sufficiency condition without assuming strict complementarity of the solution. Numerical experiments are presented that confirm the robustness and efficiency of the approach. [Copyright &y& Elsevier]

In this paper, we propose a non-monotone line search multidimensional filter-SQP method for general nonlinear programming based on the Wächter-Biegler methods for nonlinear equality constrained programming. Under mild conditions, the global convergence of the new method is proved. Furthermore, with the non-monotone technique and second order correction step, it is shown that the proposed method does not suffer from the Maratos effect, so that fast local convergence to second order sufficient local solutions is achieved. Numerical results show that the new approach is efficient. [ABSTRACT FROM AUTHOR]

Abstract: Filter methods were initially designed for nonlinear programming problems by Fletcher and Leyffer. In this paper we propose a secant algorithm with line search filter method for nonlinear equality constrained optimization. The algorithm yields the global convergence under some reasonable conditions. By using the Lagrangian function value in the filter we establish that the proposed algorithm can overcome the Maratos effect without using second order correction step, so that fast local superlinear convergence to second order sufficient local solution is achieved. The primary numerical results are presented to confirm the robustness and efficiency of our approach. [ABSTRACT FROM AUTHOR]

In this paper, we propose a nonmonotone sequential quadratic programming-filter method for solving nonlinear equality constrained optimization. This new method has more flexibility for the acceptance of the trial step and requires less computational costs compared with the monotone methods. Under reasonable conditions, we give the global convergence properties. Further, the second-order correction step and nonmonotone reduction conditions are used to overcome Maratos effect so that quadratic local convergence is achieved. The 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: 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 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 LC1 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. [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 two-piece update of projected Hessian algorithm with trust region method for solving nonlinear equality constrained optimization problems. In order to deal with large scale problems, a two-piece update of two side reduced Hessian is used to replace full Hessian matrix. By adopting the l1 penalty function as the merit function, a nonmonotonic trust region strategy is suggested which does not require the merit function to reduce its value in every iteration. The calculation of a correction step, which is necessary from theoretical point to overcome Maratos effect but sometime brings in negative results in practice, is avoided in most cases by setting a criterion to judge its necessity. The proposed algorithm which switches to nonmonotonic trust region strategy possess global convergence while maintaining one-step Q-superlinear local convergence rates if at least one of the update formula is updated at each iteration. The numerical experiment is reported to show the effectiveness of the proposed algorithms. [Copyright &y& Elsevier]

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]

We present inexact secant methods in association with line search filter technique for solving nonlinear equality constrained optimization. For large-scale applications, it is expensive to get an exact search direction, and hence we use an inexact method that finds an approximate solution satisfying some appropriate conditions. The global convergence of the proposed algorithm is established by using line search filter technique. The second-order correction step is used to overcome the Maratos effect, while the line search filter inexact secant methods have superlinear local convergence rate. Finally, the results of numerical experiments indicate that the proposed methods are efficient for the given test problems. [ABSTRACT FROM AUTHOR]