A PRECONDITIONING FRAMEWORK FOR SEQUENCES OF DIAGONALLY MODIFIED LINEAR SYSTEMS ARISING IN OPTIMIZATION
We propose a framework for building preconditioned for sequences of linear systems of the form (A + ∆ k )x k = b k , where A is symmetric positive semidefinite and ∆ k is diagonal positive semidefinite. Such sequences arise in several optimization methods, e.g., in affine-scaling methods for bound-c...
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| Vydáno v: | SIAM journal on numerical analysis Ročník 50; číslo 6; s. 3280 - 3302 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Society for Industrial and Applied Mathematics
01.01.2012
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| ISSN: | 0036-1429, 1095-7170 |
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| Abstract | We propose a framework for building preconditioned for sequences of linear systems of the form (A + ∆ k )x k = b k , where A is symmetric positive semidefinite and ∆ k is diagonal positive semidefinite. Such sequences arise in several optimization methods, e.g., in affine-scaling methods for bound-constrained convex quadratic programming and bound-constrained linear least squares, as well as in trust-region and overestimation methods for convex unconstrained optimization problems and nonlinear least squares. For all the matrices of a sequence, the preconditioners are obtained by updating any preconditioner for A available in the LDL T form. The preconditioners in the framework satisfy the natural requirement of being effective on slowly varying sequences; furthermore, under an additional property they are also able to cluster eigenvalues of the preconditioned matrix when some entries of ∆ k are sufficiently large. We present two low-cost preconditioners sharing the above-mentioned properties and evaluate them on sequences of linear systems generated by the reflective Newton method applied to bound-constrained convex quadratic programming problems and on sequences arising in solving nonlinear least-squares problems with the regularized Euclidean residual method. The results of the numerical experiments show the effectiveness of these preconditioners. |
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| AbstractList | We propose a framework for building preconditioned for sequences of linear systems of the form (A + ∆ k )x k = b k , where A is symmetric positive semidefinite and ∆ k is diagonal positive semidefinite. Such sequences arise in several optimization methods, e.g., in affine-scaling methods for bound-constrained convex quadratic programming and bound-constrained linear least squares, as well as in trust-region and overestimation methods for convex unconstrained optimization problems and nonlinear least squares. For all the matrices of a sequence, the preconditioners are obtained by updating any preconditioner for A available in the LDL T form. The preconditioners in the framework satisfy the natural requirement of being effective on slowly varying sequences; furthermore, under an additional property they are also able to cluster eigenvalues of the preconditioned matrix when some entries of ∆ k are sufficiently large. We present two low-cost preconditioners sharing the above-mentioned properties and evaluate them on sequences of linear systems generated by the reflective Newton method applied to bound-constrained convex quadratic programming problems and on sequences arising in solving nonlinear least-squares problems with the regularized Euclidean residual method. The results of the numerical experiments show the effectiveness of these preconditioners. We propose a framework for building preconditioners for sequences of linear systems of the form $(A+\Delta_k) x_k=b_k$, where $A$ is symmetric positive semidefinite and $\Delta_k$ is diagonal positive semidefinite. Such sequences arise in several optimization methods, e.g., in affine-scaling methods for bound-constrained convex quadratic programming and bound-constrained linear least squares, as well as in trust-region and overestimation methods for convex unconstrained optimization problems and nonlinear least squares. For all the matrices of a sequence, the preconditioners are obtained by updating any preconditioner for $A$ available in the $LDL^T$ form. The preconditioners in the framework satisfy the natural requirement of being effective on slowly varying sequences; furthermore, under an additional property they are also able to cluster eigenvalues of the preconditioned matrix when some entries of $\Delta_k$ are sufficiently large. We present two low-cost preconditioners sharing the above-mentioned properties and evaluate them on sequences of linear systems generated by the reflective Newton method applied to bound-constrained convex quadratic programming problems and on sequences arising in solving nonlinear least-squares problems with the regularized Euclidean residual method. The results of the numerical experiments show the effectiveness of these preconditioners. [PUBLICATION ABSTRACT] |
| Author | MORINI, BENEDETTA BELLAVIA, STEFANIA DI SERAFINO, DANIELA DE SIMONE, VALENTINA |
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| CitedBy_id | crossref_primary_10_1007_s10479_019_03254_7 crossref_primary_10_1016_j_cam_2014_02_035 crossref_primary_10_1002_nla_2054 crossref_primary_10_1002_nla_2144 crossref_primary_10_1007_s40314_021_01544_0 crossref_primary_10_1080_10556788_2014_908878 crossref_primary_10_1080_10556788_2012_762517 crossref_primary_10_1016_j_amc_2017_08_029 crossref_primary_10_1016_j_camwa_2014_03_005 crossref_primary_10_1155_2019_6301326 crossref_primary_10_1137_110840819 crossref_primary_10_1137_130947155 crossref_primary_10_1007_s10589_016_9830_4 crossref_primary_10_1080_00207160_2016_1148809 crossref_primary_10_1155_2013_767042 crossref_primary_10_3390_axioms7030044 |
| Cites_doi | 10.1007/BF01389450 10.1007/s10107-009-0286-5 10.1137/S1052623494240456 10.1002/nla.610 10.1007/s101070100263 10.1137/0904038 10.1002/nla.502 10.1016/0377-0427(88)90358-5 10.1007/s10589-011-9446-7 10.1137/100803419 10.1007/s10543-008-0206-8 10.1137/0917003 10.1137/S1052623497322735 10.1137/S1064827500371244 10.1137/080732432 10.1137/S1064827596303843 10.1007/BF01932996 10.1137/08072440X 10.1137/070708494 10.1023/A:1026089811044 10.1145/962437.962439 |
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| Snippet | We propose a framework for building preconditioned for sequences of linear systems of the form (A + ∆ k )x k = b k , where A is symmetric positive semidefinite... We propose a framework for building preconditioners for sequences of linear systems of the form $(A+\Delta_k) x_k=b_k$, where $A$ is symmetric positive... |
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| SubjectTerms | Algorithms Convex analysis Eigenvalues Factorization Hessian matrices Linear systems Mathematical sequences Matrices Methods Newtons method Numerical analysis Optimization Preconditioning Quadratic programming |
| Title | A PRECONDITIONING FRAMEWORK FOR SEQUENCES OF DIAGONALLY MODIFIED LINEAR SYSTEMS ARISING IN OPTIMIZATION |
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