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The well known constant rank constraint qualification [Math. Program. Study21:110--126, 1984] introduced by Janin for nonlinear programming has beenrecently extended to a conic context by exploiting the eigenvector structure ofthe problem. In this paper we propose a more general and geometric approach fordefining a new extension of this condition to the conic context. The mainadvantage of our approach is that we are able to recast the strong second-orderproperties of the constant rank condition in a conic context. In particular, weobtain a second-order necessary optimality condition that is stronger than theclassical one obtained under Robinson's constraint qualification, in the sensethat it holds for every Lagrange multiplier, even though our condition isindependent of Robinson's condition.

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