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It is well-known that the sequence of iterations of the composition ofprojections onto closed affine subspaces converges linearly to the projectiononto the intersection of the affine subspaces when the sum of the correspondinglinear subspaces is closed. Inspired by this, in this work, we systematicallystudy the relation between the projection onto intersection of halfspaces andhyperplanes, and the composition of projections onto halfspaces andhyperplanes. In addition, as by-products, we provide the Karush-Kuhn-Tuckerconditions for characterizing the optimal solution of convex optimization withfinitely many equality and inequality constraints in Hilbert spaces andconstruct an explicit formula for the projection onto the intersection ofhyperplane and halfspace.

Projecting onto intersections of halfspaces and hyperplanes