Sensitivity Analysis in Nonlinear Semidefinite Programming

I have been conducting research on sensitivity analysis in nonlinear semidefinite programming (SDP) for over 25 years. My journey began in 1999 with Professor Jie Sun, when we established the strong semismoothness of the metric projector over the SDP cone. By taking advantage of this property, I solved the long-standing open question of characterizing Robinson’s strong regularity of nonlinear SDP problems. Consequently, Robinson’s strong regularity for linear SDP is proven to be true if and only if the primal nondegeneracy and the dual nondegeneracy hold simutaneously. Meanwhile, by using the strong semismoothness of the metric projector over the SDP cone, together with Professor Houduo Qi, we designed a highly efficient quadratically convergent semismooth Newton method for computing the nearest correlation matrix problem in “A quadratically convergent Newton method for computing the nearest correlation matrix” (the problem comes from finance and the “NCM” term was initially introduced by late Professor Nick Higham). The next milestone is the characterization of the robust isolated calmness for a class of conic programming problems. This line of inquiry culminated in achieving a long-standing goal: demonstrating that the Aubin property is equivalent to Robinson’s strong regularity at a local optimal solution for nonlinear SDP. What follows is a brief overview of my research in this area.