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Local duality of nonlinear semidefinite programming

Local duality of nonlinear semidefinite programming
Local duality of nonlinear semidefinite programming
Recently, Chan and Sun [Chan, Z. X., D. Sun. Constraint nondegeneracy, strong regularity and nonsingularity in semidefinite programming. SIAM J. Optim.19 370--376.] reported for semidefinite programming (SDP) that the primal/dual constraint nondegeneracy is equivalent to the dual/primal strong second order sufficient condition (SSOSC). This result is responsible for a number of important results in stability analysis of SDP. In this paper, we study duality of this type in nonlinear semidefinite programming (NSDP). We introduce the dual SSOSC at a Karush-Kuhn-Tucker (KKT) triple of NSDP and study its various characterizations and relationships to the primal nondegeneracy. Although the dual SSOSC is nothing but the SSOSC for the Wolfe dual of the NSDP, it suggests new information for the primal NSDP. For example, it ensures that the inverse of the Hessian of the Lagrangian function exists at the KKT triple and the inverse is positive definite on some normal space. It also ensures the primal nondegeneracy. Some of our results generalize the corresponding classical duality results in nonlinear programming studied by Fujiwara et al. [Fujiwara, O., S.-P. Han, O. L. Mangasarian. 1984. Local duality of nonlinear programs. SIAM J. Control Optim.22 162--169]. For the convex quadratic SDP (QSDP), we have complete characterizations for the primal and dual SSOSC. Our results reveal that the nearest correlation matrix problem satisfies not only the primal and dual SSOSC but also the primal and dual nondegeneracy at its solution, suggesting that it is a well-conditioned QSDP
constraint nondegeneracy, nonlinear semidefinite programming, nonsingularity, strong second-order sufficient condition
0364-765X
124-141
Qi, Houduo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85
Qi, Houduo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85

Qi, Houduo (2009) Local duality of nonlinear semidefinite programming. Mathematics of Operations Research, 34 (1), 124-141. (doi:10.1287/moor.1080.0315).

Record type: Article

Abstract

Recently, Chan and Sun [Chan, Z. X., D. Sun. Constraint nondegeneracy, strong regularity and nonsingularity in semidefinite programming. SIAM J. Optim.19 370--376.] reported for semidefinite programming (SDP) that the primal/dual constraint nondegeneracy is equivalent to the dual/primal strong second order sufficient condition (SSOSC). This result is responsible for a number of important results in stability analysis of SDP. In this paper, we study duality of this type in nonlinear semidefinite programming (NSDP). We introduce the dual SSOSC at a Karush-Kuhn-Tucker (KKT) triple of NSDP and study its various characterizations and relationships to the primal nondegeneracy. Although the dual SSOSC is nothing but the SSOSC for the Wolfe dual of the NSDP, it suggests new information for the primal NSDP. For example, it ensures that the inverse of the Hessian of the Lagrangian function exists at the KKT triple and the inverse is positive definite on some normal space. It also ensures the primal nondegeneracy. Some of our results generalize the corresponding classical duality results in nonlinear programming studied by Fujiwara et al. [Fujiwara, O., S.-P. Han, O. L. Mangasarian. 1984. Local duality of nonlinear programs. SIAM J. Control Optim.22 162--169]. For the convex quadratic SDP (QSDP), we have complete characterizations for the primal and dual SSOSC. Our results reveal that the nearest correlation matrix problem satisfies not only the primal and dual SSOSC but also the primal and dual nondegeneracy at its solution, suggesting that it is a well-conditioned QSDP

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More information

Published date: February 2009
Keywords: constraint nondegeneracy, nonlinear semidefinite programming, nonsingularity, strong second-order sufficient condition
Organisations: Operational Research

Identifiers

Local EPrints ID: 43243
URI: http://eprints.soton.ac.uk/id/eprint/43243
ISSN: 0364-765X
PURE UUID: 88e49ebd-c7b7-43ef-b0f0-33d1a95dcd26
ORCID for Houduo Qi: ORCID iD orcid.org/0000-0003-3481-4814

Catalogue record

Date deposited: 22 Mar 2010
Last modified: 09 Jan 2022 03:17

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