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An exact duality theory for semidefinite programming based on sums of squares

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An exact duality theory for semidefinite programming based on sums of squares

Type of Publication: Journal article
Author: Schweighofer, Markus; Klep, Igor
Year of publication: 2013
Published in: Mathematics of Operations Research ; 38 (2013), 3. - pp. 569-590. - ISSN 0364-765X. - eISSN 1526-5471
DOI (citable link): https://dx.doi.org/10.1287/moor.1120.0584
Summary:
Farkas' lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry: A linear matrix inequality A(x) >_ 0 is infeasible if and only if −1 lies in the quadratic module associated to A. We also present a new exact duality theory for semidefinite programming, motivated by the real radical and sums of squares certificates from real algebraic geometry.
Subject (DDC): 510 Mathematics
Keywords: linear matrix inequality, LMI, spectrahedron, semidefinite programming, SDP, quadratic module, infeasibility, duality theory, real radical, Farkas' lemma
Bibliography of Konstanz: Yes

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SCHWEIGHOFER, Markus, Igor KLEP, 2013. An exact duality theory for semidefinite programming based on sums of squares. In: Mathematics of Operations Research. 38(3), pp. 569-590. ISSN 0364-765X. eISSN 1526-5471. Available under: doi: 10.1287/moor.1120.0584

@article{Schweighofer2013exact-24805, title={An exact duality theory for semidefinite programming based on sums of squares}, year={2013}, doi={10.1287/moor.1120.0584}, number={3}, volume={38}, issn={0364-765X}, journal={Mathematics of Operations Research}, pages={569--590}, author={Schweighofer, Markus and Klep, Igor} }

Farkas' lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry: A linear matrix inequality A(x) >_ 0 is infeasible if and only if −1 lies in the quadratic module associated to A. We also present a new exact duality theory for semidefinite programming, motivated by the real radical and sums of squares certificates from real algebraic geometry. Schweighofer, Markus An exact duality theory for semidefinite programming based on sums of squares 2013 eng Mathematics of Operations Research ; 38 (2013), 3. - S. 569-590 2013-10-11T14:29:51Z terms-of-use Klep, Igor Klep, Igor 2013-10-11T14:29:51Z Schweighofer, Markus

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