Computations of the limiting (Mordukhovich) coderivative of the metric projection onto non-polyhedral convexs sets

The limiting (Mordukhovich) coderivative of the metric projection onto a convex set S has played a central role in variational analysis, particularly in the study of the Aubin property. However, for non-polyhedral sets S, it was not known whether explicit formulas for these coderivatives could be computed. This situation began to change in 2008, when, together with Professor Jiri Outrata, we successfully derived explicit coderivative formulas for the metric projection onto the second-order cone. Our results were published in Jiri Outrata and Defeng Sun, “On the coderivative of the projection operator onto the second order cone”, Set-Valued Analysis 16 (2008) 999–1014. These coderivative formulas have since found important applications. For instance, in 2025, Liang Chen, Ruoning Chen, Defeng Sun, and Junyuan Zhu used them in their paper, “Aubin property and strong regularity are equivalent for nonlinear second-order cone programming”, published in SIAM Journal on Optimization 35:2 (2025) 712–738. In this work, they established the equivalence between the Aubin property and Robinson’s strong regularity for nonlinear second-order cone programming. See the Flowchart of the Proof. Further progress was made in 2014, when Chao Ding, Defeng Sun, and Jane Ye derived explicit formulas for the metric projection onto the cone of symmetric and positive semidefinite matrices—an important non-polyhedral cone in semidefinite programming. Their results appeared in “First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints”, Mathematical Programming 147 (2014) 539-579.