Matching-based preprocessing algorithms to the solution of saddle-point problems in large-scale nonconvex interior-point optimization

Interior-point methods are among the most efficient approaches for solving large-scale nonlinear programming problems. At the core of these methods, highly ill-conditioned symmetric saddle-point problems have to be solved. We present combinatorial methods to preprocess these matrices in order to est...

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Veröffentlicht in:	Computational optimization and applications Jg. 36; H. 2-3; S. 321 - 341
Hauptverfasser:	Schenk, Olaf, Wächter, Andreas, Hagemann, Michael
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer Nature B.V 01.04.2007
Schlagworte:	Algorithms Combinatorics Linear algebra Mathematical models Methods Nonlinear programming Optimization Optimization algorithms Partial differential equations Studies Variables
ISSN:	0926-6003, 1573-2894
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Abstract	Interior-point methods are among the most efficient approaches for solving large-scale nonlinear programming problems. At the core of these methods, highly ill-conditioned symmetric saddle-point problems have to be solved. We present combinatorial methods to preprocess these matrices in order to establish more favorable numerical properties for the subsequent factorization. Our approach is based on symmetric weighted matchings and is used in a sparse direct LDL T factorization method where the pivoting is restricted to static supernode data structures. In addition, we will dynamically expand the supernode data structure in cases where additional fill-in helps to select better numerical pivot elements. This technique can be seen as an alternative to the more traditional threshold pivoting techniques. We demonstrate the competitiveness of this approach within an interior-point method on a large set of test problems from the CUTE and COPS sets, as well as large optimal control problems based on partial differential equations. The largest nonlinear optimization problem solved has more than 12 million variables and 6 million constraints. [PUBLICATION ABSTRACT]
AbstractList	Interior-point methods are among the most efficient approaches for solving large-scale nonlinear programming problems. At the core of these methods, highly ill-conditioned symmetric saddle-point problems have to be solved. We present combinatorial methods to preprocess these matrices in order to establish more favorable numerical properties for the subsequent factorization. Our approach is based on symmetric weighted matchings and is used in a sparse direct LDL T factorization method where the pivoting is restricted to static supernode data structures. In addition, we will dynamically expand the supernode data structure in cases where additional fill-in helps to select better numerical pivot elements. This technique can be seen as an alternative to the more traditional threshold pivoting techniques. We demonstrate the competitiveness of this approach within an interior-point method on a large set of test problems from the CUTE and COPS sets, as well as large optimal control problems based on partial differential equations. The largest nonlinear optimization problem solved has more than 12 million variables and 6 million constraints. [PUBLICATION ABSTRACT] Interior-point methods are among the most efficient approaches for solving large-scale nonlinear programming problems. At the core of these methods, highly ill-conditioned symmetric saddle-point problems have to be solved. We present combinatorial methods to preprocess these matrices in order to establish more favorable numerical properties for the subsequent factorization. Our approach is based on symmetric weighted matchings and is used in a sparse direct LDL factorization method where the pivoting is restricted to static supernode data structures. In addition, we will dynamically expand the supernode data structure in cases where additional fill-in helps to select better numerical pivot elements. This technique can be seen as an alternative to the more traditional threshold pivoting techniques. We demonstrate the competitiveness of this approach within an interior-point method on a large set of test problems from the CUTE and COPS sets, as well as large optimal control problems based on partial differential equations. The largest nonlinear optimization problem solved has more than 12 million variables and 6 million constraints.
Author	Hagemann, Michael Schenk, Olaf Wächter, Andreas
Author_xml	– sequence: 1 givenname: Olaf surname: Schenk fullname: Schenk, Olaf – sequence: 2 givenname: Andreas surname: Wächter fullname: Wächter, Andreas – sequence: 3 givenname: Michael surname: Hagemann fullname: Hagemann, Michael
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Snippet	Interior-point methods are among the most efficient approaches for solving large-scale nonlinear programming problems. At the core of these methods, highly...
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Title	Matching-based preprocessing algorithms to the solution of saddle-point problems in large-scale nonconvex interior-point optimization
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