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

Publikationstyp: Zeitschriftenartikel
Autor/innen: Schweighofer, Markus; Klep, Igor
Erscheinungsjahr: 2013
Erschienen in: Mathematics of Operations Research ; 38 (2013), 3. - S. 569-590. - ISSN 0364-765X. - eISSN 1526-5471
DOI (zitierfähiger Link): https://dx.doi.org/10.1287/moor.1120.0584
Zusammenfassung:
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.
Fachgebiet (DDC): 510 Mathematik
Schlagwörter: linear matrix inequality, LMI, spectrahedron, semidefinite programming, SDP, quadratic module, infeasibility, duality theory, real radical, Farkas' lemma
Link zur Lizenz: Deposit-Lizenz
Universitätsbibliographie: Ja

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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

@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. deposit-license Schweighofer, Markus 2013 An exact duality theory for semidefinite programming based on sums of squares eng Mathematics of Operations Research ; 38 (2013), 3. - S. 569-590 2013-10-11T14:29:51Z Klep, Igor Klep, Igor 2013-10-11T14:29:51Z Schweighofer, Markus

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