  In the last two decades, the sequential optimality conditions, which do not
require constraint qualifications and allow improvement on the convergence
assumptions of algorithms, had been considered in the literature. It includes
the work by Andreani et al. (2017), with a sequential optimality condition for
nonlinear programming, that uses the second-order information of the problem.
More recently, Fukuda et al. (2023) analyzed the conditions that use
second-order information, in particular for nonlinear second-order cone
programming problems (SOCP). However, such optimality conditions were not
defined explicitly. In this paper, we propose an explicit definition of
approximate-Karush-Kuhn-Tucker 2 (AKKT2) conditions for SOCPs. We prove that
the proposed AKKT2 conditions are satisfied at local optimal points of the SOCP
without any constraint qualification. We also present two algorithms that are
based on augmented Lagrangian and sequential quadratic programming methods and
show their global convergence to points satisfying the proposed conditions.
