In this paper, we give necessary and sufficient optimality conditions which are easy verified for the local solution of Celis-Dennis-Tapia subproblem (CDT subproblem) where the Hessian at this local solution has one negative eigenvalue. If CDT subproblem has no global solution with Hessian of Lagrangian positive semi-definite, the Hessian of Lagrangian has at least one negative eigenvalue. It is very important to investigate all hte stationary points of Lagrangian dual function and to characterize the local solutions. We also discuss the gap between these two conditions.
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