Quadratic programs with quadratic constraint : characterization of KKT points and equivalencewith an unconstrained problem

In this paper we consider the problem of minimizing a quadratic function with a quadratic constraint. We point out some new properties of the problem. In particular, in the rst part of the paper, we show that (i) the number of values of the objective function at KKT points is bounded by 3n + 1 where n is the dimension of the problem; (ii) given a KKT point that is not a global minimizer, it is immediate to nd a \better" feasible point; (iii) strict complementarity holds at the local-nonglobal minimum point. In the second part, we show that the original constrained problem is equivalent to the unconstrained minimization of a piecewise quartic exact merit function. Using the unconstrained formulation we give, in the nonconvex case, a new second order necessary condition for global minimimum points. A possible algorithmic application of the preceding results is brie y outlined.