First- and second-order optimality conditions for second-order cone and semidefinite programming under a constant rank condition

The well known constant rank constraint qualification [Math. Program. Study 21:110-126, 1984] introduced by Janin for nonlinear programming has been recently extended to a conic context by exploiting the eigenvector structure of the problem. In this paper we propose a more general and geometric approach for defining a new extension of this condition to the conic context. The main advantage of our approach is that we are able to recast the strong second-order properties of the constant rank condition in a conic context. In particular, we obtain a second-order necessary optimality condition that is stronger than the classical one obtained under Robinson's constraint qualification, in the sense that it holds for every Lagrange multiplier, even though our condition is independent of Robinson's condition.

Automated summary

In this paper, a general and geometric approach is proposed for defining a new extension of the constant rank condition to the conic context. The main advantage is that the strong second-order properties of the constant rank condition can be recast in a conic context. Specifically, a second-order necessary optimality condition is obtained that is stronger than the classical one obtained under Robinson's constraint qualification.