On an Extension of Condition Number Theory to Non-Conic Convex Optimization
The purpose of this paper is to extend, as much as possible, the modern theory of condition numbers for conic convex optimization: z* := minz ctx s.t. Ax - b Cy C Cx , to the more general non-conic format: z* := minx ctx (GPd) s.t. Ax-b E Cy X P, where P is any closed convex set, not necessarily a c...
| Main Authors: | , |
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| Format: | Working Paper |
| Language: | en_US |
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Massachusetts Institute of Technology, Operations Research Center
2004
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| Online Access: | http://hdl.handle.net/1721.1/5404 |
| _version_ | 1826203303364526080 |
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| author | Freund, Robert M. Ordóñez, Fernando, 1970- |
| author_facet | Freund, Robert M. Ordóñez, Fernando, 1970- |
| author_sort | Freund, Robert M. |
| collection | MIT |
| description | The purpose of this paper is to extend, as much as possible, the modern theory of condition numbers for conic convex optimization: z* := minz ctx s.t. Ax - b Cy C Cx , to the more general non-conic format: z* := minx ctx (GPd) s.t. Ax-b E Cy X P, where P is any closed convex set, not necessarily a cone, which we call the groundset. Although any convex problem can be transformed to conic form, such transformations are neither unique nor natural given the natural description of many problems, thereby diminishing the relevance of data-based condition number theory. Herein we extend the modern theory of condition numbers to the problem format (GPd). As a byproduct, we are able to state and prove natural extensions of many theorems from the conic-based theory of condition numbers to this broader problem format. |
| first_indexed | 2024-09-23T12:34:30Z |
| format | Working Paper |
| id | mit-1721.1/5404 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T12:34:30Z |
| publishDate | 2004 |
| publisher | Massachusetts Institute of Technology, Operations Research Center |
| record_format | dspace |
| spelling | mit-1721.1/54042019-04-10T10:36:20Z On an Extension of Condition Number Theory to Non-Conic Convex Optimization Freund, Robert M. Ordóñez, Fernando, 1970- Condition number, convex optimization, conic optimization, duality, sensitivity analysis, perturbation theory. The purpose of this paper is to extend, as much as possible, the modern theory of condition numbers for conic convex optimization: z* := minz ctx s.t. Ax - b Cy C Cx , to the more general non-conic format: z* := minx ctx (GPd) s.t. Ax-b E Cy X P, where P is any closed convex set, not necessarily a cone, which we call the groundset. Although any convex problem can be transformed to conic form, such transformations are neither unique nor natural given the natural description of many problems, thereby diminishing the relevance of data-based condition number theory. Herein we extend the modern theory of condition numbers to the problem format (GPd). As a byproduct, we are able to state and prove natural extensions of many theorems from the conic-based theory of condition numbers to this broader problem format. 2004-06-01T16:43:00Z 2004-06-01T16:43:00Z 2003-02 Working Paper http://hdl.handle.net/1721.1/5404 en_US Operations Research Center Working Paper;OR 365-03 2161257 bytes application/pdf application/pdf Massachusetts Institute of Technology, Operations Research Center |
| spellingShingle | Condition number, convex optimization, conic optimization, duality, sensitivity analysis, perturbation theory. Freund, Robert M. Ordóñez, Fernando, 1970- On an Extension of Condition Number Theory to Non-Conic Convex Optimization |
| title | On an Extension of Condition Number Theory to Non-Conic Convex Optimization |
| title_full | On an Extension of Condition Number Theory to Non-Conic Convex Optimization |
| title_fullStr | On an Extension of Condition Number Theory to Non-Conic Convex Optimization |
| title_full_unstemmed | On an Extension of Condition Number Theory to Non-Conic Convex Optimization |
| title_short | On an Extension of Condition Number Theory to Non-Conic Convex Optimization |
| title_sort | on an extension of condition number theory to non conic convex optimization |
| topic | Condition number, convex optimization, conic optimization, duality, sensitivity analysis, perturbation theory. |
| url | http://hdl.handle.net/1721.1/5404 |
| work_keys_str_mv | AT freundrobertm onanextensionofconditionnumbertheorytononconicconvexoptimization AT ordonezfernando1970 onanextensionofconditionnumbertheorytononconicconvexoptimization |