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Local Duality of Nonlinear Semidefinite Programming

School of Mathematics, The University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom
hdqi{at}soton.ac.uk

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.

Key Words: nonlinear semidefinite programming; strong second-order sufficient condition; constraint nondegeneracy; nonsingularity
History: Received: April 25, 2007; revision received: January 6, 2008;