Mathematical Programming: Series A and B
Convergence analysis of some algorithms for solving nonsmooth equations
Mathematics of Operations Research
Complementarity and nondegeneracy in semidefinite programming
Mathematical Programming: Series A and B
First and second order analysis of nonlinear semidefinite programs
Mathematical Programming: Series A and B
Sensitivity Analysis of Optimization Problems Under Second Order Regular Constraints
Mathematics of Operations Research
Semidefinite Programs: New Search Directions, Smoothing-Type Methods, and Numerical Results
SIAM Journal on Optimization
SIAM Journal on Optimization
Second Order Optimality Conditions Based on Parabolic Second Order Tangent Sets
SIAM Journal on Optimization
Semismooth Matrix-Valued Functions
Mathematics of Operations Research
Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems
Mathematics of Operations Research
Constraint nondegeneracy in variational analysis
Mathematics of Operations Research
Mathematics of Operations Research
An inexact primal–dual path following algorithm for convex quadratic SDP
Mathematical Programming: Series A and B
Constraint Nondegeneracy, Strong Regularity, and Nonsingularity in Semidefinite Programming
SIAM Journal on Optimization
On the convergence of augmented Lagrangian methods for nonlinear semidefinite programming
Journal of Global Optimization
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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.