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Volume 20, Issue 3
An Interior Trust Region Algorithm for Nonlinear Minimization with Linear Constraints

Jian-Guo Liu

J. Comp. Math., 20 (2002), pp. 225-244.

Published online: 2002-06

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  • Abstract

An interior trust-region-based algorithm for linearly constrained minimization problems is proposed and analyzed. This algorithm is similar to trust region algorithms for unconstrained minimization: a trust region subproblem on a subspace is solved in each iteration. We establish that the proposed algorithm has convergence properties analogous to those of the trust region algorithms for unconstrained minimization. Namely, every limit point of the generated sequence satisfies the Krush-Kuhn-Tucker (KKT) conditions and at least one limit point satisfies second order necessary optimality conditions. In addition, if one limit point is a strong local minimizer and the Hessian is Lipschitz continuous in a neighborbood of that point, then the generated sequence converges globally to that point in the rate of at least 2-step quadratic. We are mainly concerned with the theoretical properties of the algorithm in this paper. Implementation issues and adaptation to large-scale problems will be addressed in a future report.

  • Keywords

Nonlinear programming, Linear constraints, Trust region algorithms, Newton methods, Interior algorithms, Quadratic convergence.

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COPYRIGHT: © Global Science Press

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@Article{JCM-20-225, author = {Liu , Jian-Guo}, title = {An Interior Trust Region Algorithm for Nonlinear Minimization with Linear Constraints}, journal = {Journal of Computational Mathematics}, year = {2002}, volume = {20}, number = {3}, pages = {225--244}, abstract = {

An interior trust-region-based algorithm for linearly constrained minimization problems is proposed and analyzed. This algorithm is similar to trust region algorithms for unconstrained minimization: a trust region subproblem on a subspace is solved in each iteration. We establish that the proposed algorithm has convergence properties analogous to those of the trust region algorithms for unconstrained minimization. Namely, every limit point of the generated sequence satisfies the Krush-Kuhn-Tucker (KKT) conditions and at least one limit point satisfies second order necessary optimality conditions. In addition, if one limit point is a strong local minimizer and the Hessian is Lipschitz continuous in a neighborbood of that point, then the generated sequence converges globally to that point in the rate of at least 2-step quadratic. We are mainly concerned with the theoretical properties of the algorithm in this paper. Implementation issues and adaptation to large-scale problems will be addressed in a future report.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8913.html} }
TY - JOUR T1 - An Interior Trust Region Algorithm for Nonlinear Minimization with Linear Constraints AU - Liu , Jian-Guo JO - Journal of Computational Mathematics VL - 3 SP - 225 EP - 244 PY - 2002 DA - 2002/06 SN - 20 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8913.html KW - Nonlinear programming, Linear constraints, Trust region algorithms, Newton methods, Interior algorithms, Quadratic convergence. AB -

An interior trust-region-based algorithm for linearly constrained minimization problems is proposed and analyzed. This algorithm is similar to trust region algorithms for unconstrained minimization: a trust region subproblem on a subspace is solved in each iteration. We establish that the proposed algorithm has convergence properties analogous to those of the trust region algorithms for unconstrained minimization. Namely, every limit point of the generated sequence satisfies the Krush-Kuhn-Tucker (KKT) conditions and at least one limit point satisfies second order necessary optimality conditions. In addition, if one limit point is a strong local minimizer and the Hessian is Lipschitz continuous in a neighborbood of that point, then the generated sequence converges globally to that point in the rate of at least 2-step quadratic. We are mainly concerned with the theoretical properties of the algorithm in this paper. Implementation issues and adaptation to large-scale problems will be addressed in a future report.

Liu , Jian-Guo. (2002). An Interior Trust Region Algorithm for Nonlinear Minimization with Linear Constraints. Journal of Computational Mathematics. 20 (3). 225-244. doi:
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