"Perturbation analysis of semidefinite programming problems"
The theory of second order optimality conditions deals with conditions involving the Hessian of the Lagrangian function along critical directions, as for nonlinear programming problems. However, they involve also a measure of the curvature of the set of semi-definite positive matrices along such directions. This additional term, originally introduced by Kawasaki for some semi infinite programming problems, can be computed in the case of semidefinite programming problems, in terms of the eigenvalue and eigenvectors of the matrix.
A specific aspect of the theory is that no-gap conditions are obtained, in the sense that a characterization of optimality up to the second order is obtained or, equivalently, a characterization of the quadratic growth condition can be obtained in terms of second order derivatives of data. This is due to the fact that a certain hypothesis of second order regularity is satisfied. This hypothesis , roughly speaking, states that feasible paths are "close" to "parabolic" paths.
The perturbation analysis, based on upper and lower estimates of the value of the perturbed problem, allows to compute expansions of the value function and solution, under hypotheses that are close to those made in the theory of perturbed nonlinear programs.
References
[1] J.F. Bonnans, R. Cominetti and A. Shapiro (1999),
"Second order optimality conditions based on
parabolic second order tangent sets",
SIAM J. Optimization 9, to appear.
[2] J.F. Bonnans, R. Cominetti and A. Shapiro (1999),
"Sensitivity analysis of optimization problems under
second order regular constraints",
Mathematics of Operations Research, to appear.
helmberg@zib.de
Last Update: October 1, 1998