On the extensions of Frank-Wolfe theorem

In this paper we consider optimization problems defined by a quadraticobjective function and a finite number of quadratic inequality constraints.Given that the objective function is bounded over the feasible set, we presenta comprehensive study of the conditions under which the optimal solution setis nonempty, thus extending the so-called Frank-Wolfe theorem. In particular,we first prove a general continuity result for the solution set defined by asystem of convex quadratic inequalities. This result implies immediatelythat the optimal solution set of the aforementioned problem is nonempty whenall the quadratic functions involved are convex. In the absence of theconvexity of the objective function, we give examples showing that the optimalsolution set may be empty either when there are two or more convex quadraticconstraints, or when the Hessian of the objective function has two or morenegative eigenvalues. In the case when there exists only one convex quadraticinequality constraint (together with other linear constraints), or when theconstraint functions are all convex quadratic and the objective function isquasi-convex (thus allowing one negative eigenvalue in its Hessian matrix), weprove that the optimal solution set is nonempty.