Strongly stable C-stationary points for mathematical programs with complementarity constraints (Q2230944)

The authors prove a topological and an equivalent algebraic characterization of the strong stability of a C-stationary point of mathematical programs with complementarity constraints. They adapt the concept of strong stability, starting from the standard nonlinear optimization, which includes the local uniqueness existence and continuous dependence of a solution for each sufficiently small perturbed problem where perturbations up to second order are allowed. For example, the set of Lagrange vectors is frequently not convex in this framework, which arises the need of replacing this concept by a more useful one in characterizing strongly stable C-stationary points. Then, the authors introduce the set of basic Lagrange vectors, which plays a similar role to that of the set of extreme points. A necessary second order condition, called Condition \(C^{\ast}\), for the strong stability of a C-stationary point is introduced. The case of convexity of the set of Lagrange vectors is studied, proving that it is necessary for the strong stability of C-stationary points under two additional assumptions, denoted by A1 and A2. An overview on some existing concepts of stationarity for mathematical programs with complementarity constraints are finally mentioned, which opens the need of further research on characterization of strong stability in each case.