First-order Optimality Conditions for Mathematical Programs with Second-order Cone Complementarity

In this paper we consider a mathematical program with second-order cone complementarity constraints (SOCMPCC). The SOCMPCC generalizes the mathematical program with complementarity constraints (MPCC) in replacing the set of nonnegative reals by second-order cones. There are difficulties in applying the classical Karush–Kuhn–Tucker (KKT) condition to the SOCMPCC directly since the usual constraint qualification such as Robinson’s constraint qualification never holds if it is considered as an optimization problem with a convex cone constraint. Using various reformulations and recent results on the exact formula for the proximal/regular and limiting normal cone, we derive necessary optimality conditions in the forms of the strong, Mordukhovich, and Clarke (resp., S, M, and C) stationary conditions under certain constraint qualifications. We also show that, unlike the MPCC, the classical KKT condition of the SOCMPCC is in general not equivalent to the S-stationary condition unless the dimension of each second-order cone is not more than 2. Finally, we show that reformulating an MPCC as an SOCMPCC produces new and weaker necessary optimality conditions.