Stationary index and orientation of equality constrained multiparametric nonlinear programs

In this paper, the authors study a local property of the zero set of a differentiable map . They prove that, under a regular value condition, for each, there exists a neighborhood U of x and a sign such that for all permutation of degree (n+d), where p is a d-dimensional parametrization parameter vector of the zero set in an open subset V or R ^d and . This result naturally leads to an index theory. The authors show a local property of the change of the Morse index and the orientation of critical point set w.r.t. the multiparametric function . Finally, they discuss the change of the stationary index of the equality constrained multiparametric nonlinear programs.