On Mathematical Programs with Second-Order Cone Complementarity Constraints: Necessary Optimality Conditions and Constraint Qualifications

Jane Ye, Jinchuan Zhou

In this talk 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 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- (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 two. Moreover we provide some verifiable constraint qualifications for the M-stationary condition to hold at a local minimum.