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The C-Index: A New Stability Concept for Quadratic Programs with Complementarity Constraints

We introduce nondegeneracy and the C-index for C-stationary points of a QPCC, that is, for a mathematical program with a quadratic objective function and linear complementarity constraints. The C-index characterizes the qualitative local behavior of a QPCC around a nondegenerate C-stationary point. The article focuses on the structure of the C-stationary set of QPCCs depending on a real parameter. We show that, for generic QPCC data, the C-index changes exactly at turning points of the C-stationary set, and that it changes by exactly one. 
 
To illustrate this concept, we introduce and analyze two homotopy methods for finding C-stationary points. Numerical results illustrate that, for randomly generated test problems, the two homotopy methods very often identify B-stationary points.