Dualities in convex algebraic geometry
Convex algebraic geometry concerns the interplay between optimization theory and real algebraic geometry. Its objects of study include convex semialgebraic sets that arise in semidefinite programming and from sums of squares. This article compares three notions of duality that are relevant in these...
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| Format: | Article |
| Language: | English |
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Sapienza Università Editrice
2010-01-01
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| Series: | Rendiconti di Matematica e delle Sue Applicazioni |
| Subjects: | |
| Online Access: | https://www1.mat.uniroma1.it/ricerca/rendiconti/ARCHIVIO/2010(3-4)/285-327.pdf |
| _version_ | 1819081574043353088 |
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| author | Philipp Rostalski Bernd Sturmfels |
| author_facet | Philipp Rostalski Bernd Sturmfels |
| author_sort | Philipp Rostalski |
| collection | DOAJ |
| description | Convex algebraic geometry concerns the interplay between optimization
theory and real algebraic geometry. Its objects of study include convex semialgebraic sets that arise in semidefinite programming and from sums of squares. This article compares three notions of duality that are relevant in these contexts: duality of convex bodies, duality of projective varieties, and the Karush-Kuhn-Tucker conditions derived from Lagrange duality. We show that the optimal value of a polynomial program is an algebraic function whose minimal polynomial is expressed by the hypersurface projectively dual to the constraint set. We give an exposition of recent results on the boundary structure of the convex hull of a compact variety, we contrast this to Lasserre’s representation as a spectrahedral shadow, and we explore the geometric underpinnings of semidefinite programming duality. |
| first_indexed | 2024-12-21T20:02:56Z |
| format | Article |
| id | doaj.art-e5a6dfbbdc2342ee9bc6e433f7c18ced |
| institution | Directory Open Access Journal |
| issn | 1120-7183 2532-3350 |
| language | English |
| last_indexed | 2024-12-21T20:02:56Z |
| publishDate | 2010-01-01 |
| publisher | Sapienza Università Editrice |
| record_format | Article |
| series | Rendiconti di Matematica e delle Sue Applicazioni |
| spelling | doaj.art-e5a6dfbbdc2342ee9bc6e433f7c18ced2022-12-21T18:51:55ZengSapienza Università EditriceRendiconti di Matematica e delle Sue Applicazioni1120-71832532-33502010-01-01303-4285327Dualities in convex algebraic geometryPhilipp Rostalski0Bernd Sturmfels1University of CaliforniaUniversity of CaliforniaConvex algebraic geometry concerns the interplay between optimization theory and real algebraic geometry. Its objects of study include convex semialgebraic sets that arise in semidefinite programming and from sums of squares. This article compares three notions of duality that are relevant in these contexts: duality of convex bodies, duality of projective varieties, and the Karush-Kuhn-Tucker conditions derived from Lagrange duality. We show that the optimal value of a polynomial program is an algebraic function whose minimal polynomial is expressed by the hypersurface projectively dual to the constraint set. We give an exposition of recent results on the boundary structure of the convex hull of a compact variety, we contrast this to Lasserre’s representation as a spectrahedral shadow, and we explore the geometric underpinnings of semidefinite programming duality.https://www1.mat.uniroma1.it/ricerca/rendiconti/ARCHIVIO/2010(3-4)/285-327.pdfoptimizationdualitysemidefinite programmingspectrahedronconvexityreal algebraic geometry |
| spellingShingle | Philipp Rostalski Bernd Sturmfels Dualities in convex algebraic geometry Rendiconti di Matematica e delle Sue Applicazioni optimization duality semidefinite programming spectrahedron convexity real algebraic geometry |
| title | Dualities in convex algebraic geometry |
| title_full | Dualities in convex algebraic geometry |
| title_fullStr | Dualities in convex algebraic geometry |
| title_full_unstemmed | Dualities in convex algebraic geometry |
| title_short | Dualities in convex algebraic geometry |
| title_sort | dualities in convex algebraic geometry |
| topic | optimization duality semidefinite programming spectrahedron convexity real algebraic geometry |
| url | https://www1.mat.uniroma1.it/ricerca/rendiconti/ARCHIVIO/2010(3-4)/285-327.pdf |
| work_keys_str_mv | AT philipprostalski dualitiesinconvexalgebraicgeometry AT berndsturmfels dualitiesinconvexalgebraicgeometry |