  The uniform quadratic optimizatin problem (UQ) is a nonconvex quadratic
constrained quadratic programming (QCQP) sharing the same Hessian matrix. Based
on the second-order cone programming (SOCP) relaxation, we establish a new
sufficient condition to guarantee strong duality for (UQ) and then extend it to
(QCQP), which not only covers several well-known results in literature but also
partially gives answers to a few open questions. For convex constrained
nonconvex (UQ), we propose an improved approximation algorithm based on (SOCP).
Our approximation bound is dimensional independent. As an application, we
establish the first approximation bound for the problem of finding the
Chebyshev center of the intersection of several balls.
