First-Order Optimality Conditions for Mathematical Programs with Second-Order Cone Complementarity Constraints

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 a second-order cone. We show that if the SOCMPCC is considered as an optimization problem with convex cone constraints, then Robinson’s constraint qualification never holds. We derive explicit formulas for the regular and limiting normal cone of the second-order complementarity cone. Using these formulas we derive explicit expressions for the strong-, Mordukhovichand Clarke(S-, Mand C-) stationary conditions and show that they are necessary optimality conditions under certain constraint qualifications. We have also shown that the classical KKT condition is in general not equivalent to the S-stationary condition unless the dimension of each second-order cone is not more than 2. Moreover we show that reformulating an MPCC as an SOCMPCC produces new and weaker necessary optimality conditions.