A Family of Trust-Region-Based Algorithms for Unconstrained Minimization with Strong Global Convergence Properties
This paper has two aims: to exhibit very general conditions under which members of a broad class of unconstrained minimization algorithms are globally convergent in a strong sense, and to propose several new algorithms that use second derivative information and achieve such convergence. In the first...
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| Veröffentlicht in: | SIAM journal on numerical analysis 1985-02, Vol.22 (1), p.47-67 |
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| container_title | SIAM journal on numerical analysis |
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| creator | Shultz, Gerald A. Schnabel, Robert B. Byrd, Richard H. |
| description | This paper has two aims: to exhibit very general conditions under which members of a broad class of unconstrained minimization algorithms are globally convergent in a strong sense, and to propose several new algorithms that use second derivative information and achieve such convergence. In the first part of the paper we present a general trust-region-based algorithm schema that includes an undefined step selection strategy. We give general conditions on this step selection strategy under which limit points of the algorithm will satisfy first and second order necessary conditions for unconstrained minimization. Our algorithm schema is sufficiently broad to include line search algorithms as well. Next, we show that a wide range of step selection strategies satisfy the requirements of our convergence theory. This leads us to propose several new algorithms that use second derivative information and achieve strong global convergence, including an indefinite line search algorithm, several indefinite dogleg algorithms, and a modified "optimal-step" algorithm. Finally, we propose an implementation of one such indefinite dogleg algorithm. |
| doi_str_mv | 10.1137/0722003 |
| format | Article |
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In the first part of the paper we present a general trust-region-based algorithm schema that includes an undefined step selection strategy. We give general conditions on this step selection strategy under which limit points of the algorithm will satisfy first and second order necessary conditions for unconstrained minimization. Our algorithm schema is sufficiently broad to include line search algorithms as well. Next, we show that a wide range of step selection strategies satisfy the requirements of our convergence theory. This leads us to propose several new algorithms that use second derivative information and achieve strong global convergence, including an indefinite line search algorithm, several indefinite dogleg algorithms, and a modified "optimal-step" algorithm. 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Central China</collection><collection>Engineering Collection</collection><collection>Environmental Science Collection</collection><collection>ProQuest Central Basic</collection><jtitle>SIAM journal on numerical analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Shultz, Gerald A.</au><au>Schnabel, Robert B.</au><au>Byrd, Richard H.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A Family of Trust-Region-Based Algorithms for Unconstrained Minimization with Strong Global Convergence Properties</atitle><jtitle>SIAM journal on numerical analysis</jtitle><date>1985-02-01</date><risdate>1985</risdate><volume>22</volume><issue>1</issue><spage>47</spage><epage>67</epage><pages>47-67</pages><issn>0036-1429</issn><eissn>1095-7170</eissn><coden>SJNAEQ</coden><abstract>This paper has two aims: to exhibit very general conditions under which members of a broad class of unconstrained minimization algorithms are globally convergent in a strong sense, and to propose several new algorithms that use second derivative information and achieve such convergence. In the first part of the paper we present a general trust-region-based algorithm schema that includes an undefined step selection strategy. We give general conditions on this step selection strategy under which limit points of the algorithm will satisfy first and second order necessary conditions for unconstrained minimization. Our algorithm schema is sufficiently broad to include line search algorithms as well. Next, we show that a wide range of step selection strategies satisfy the requirements of our convergence theory. This leads us to propose several new algorithms that use second derivative information and achieve strong global convergence, including an indefinite line search algorithm, several indefinite dogleg algorithms, and a modified "optimal-step" algorithm. Finally, we propose an implementation of one such indefinite dogleg algorithm.</abstract><cop>Philadelphia, PA</cop><pub>Society for Industrial and Applied Mathematics</pub><doi>10.1137/0722003</doi><tpages>21</tpages></addata></record> |
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| identifier | ISSN: 0036-1429 |
| ispartof | SIAM journal on numerical analysis, 1985-02, Vol.22 (1), p.47-67 |
| issn | 0036-1429 1095-7170 |
| language | eng |
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| source | SIAM Journals; JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing |
| subjects | Algorithms Arithmetic Cholesky factorization Curvature Eigenvalues Eigenvectors Exact sciences and technology Factorization Mathematics Necessary conditions Numerical analysis Numerical analysis in abstract spaces Numerical analysis. Scientific computation Perceptron convergence procedure Sciences and techniques of general use |
| title | A Family of Trust-Region-Based Algorithms for Unconstrained Minimization with Strong Global Convergence Properties |
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