  This paper solves a fundamental open problem in variational analysis on the
equivalence between the Aubin property and the strong regularity for nonlinear
second-order cone programming (SOCP) at a locally optimal solution. We achieve
this by introducing a reduction approach to the Aubin property characterized by
the Mordukhovich criterion and a lemma of alternative choices on cones to
replace the S-lemma used in Outrata and Ram\'irez [SIAM J. Optim. 21 (2011)
789-823] and Opazo, Outrata, and Ram\'irez [SIAM J. Optim. 27 (2017)
2141-2151], where the same SOCP was considered under the strict complementarity
condition except for possibly only one block of constraints. As a byproduct, we
also offer a new approach to the well-known result of Dontchev and Rockafellar
[SIAM J. Optim. 6 (1996) 1087-1105] on the equivalence of the two concepts in
conventional nonlinear programming.
