Solution of a non-linear programming problem with quadratic functions

A problem of finding a vector of maximum length in a set determined by the intersection of a finite collection of balls is studied. Sufficient conditions for the problem to be solvable by non-combinatoric methods are formulated. In this case the ball problem can be reduced to a special convex programming problem. Some examples of the problem with violation of the sufficient conditions are considered. In the general case of the ball problem a branch-and-bound method is proposed.