  It is well-known that the sequence of iterations of the composition of
projections onto closed affine subspaces converges linearly to the projection
onto the intersection of the affine subspaces when the sum of the corresponding
linear subspaces is closed. Inspired by this, in this work, we systematically
study the relation between the projection onto intersection of halfspaces and
hyperplanes, and the composition of projections onto halfspaces and
hyperplanes. In addition, as by-products, we provide the Karush-Kuhn-Tucker
conditions for characterizing the optimal solution of convex optimization with
finitely many equality and inequality constraints in Hilbert spaces and
construct an explicit formula for the projection onto the intersection of
hyperplane and halfspace.
