  We study the space of pseudo-holomorphic spheres in compact symplectic
manifolds with convex boundary. We show that the theory of Gromov-Witten
invariants can be extended to the class of semi-positive manifolds with convex
boundary. This leads to a deformation of intersection products on the absolute
and relative singular homologies. As a result, absolute and relative quantum
homology algebras are defined analogously to the case of closed symplectic
manifolds. In addition, we prove the Poincar\'{e}-Lefschetz duality for the
absolute and relative quantum homology algebras.
