  This paper studies the polynomial optimization problem whose feasible set is
a union of several basic closed semialgebraic sets. We propose a unified
hierarchy of Moment-SOS relaxations to solve it globally. Under some
assumptions, we prove the asymptotic or finite convergence of the unified
hierarchy. Special properties for the univariate case are discussed.The
application for computing $(p,q)$-norms of matrices is also presented.
