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This paper improves the algorithms based on supporting halfspaces andquadratic programming for convex set intersection problems in our earlier paperin several directions. First, we give conditions so that much smaller quadraticprograms (QPs) and approximate projections arising from partially solving theQPs are sufficient for multiple-term superlinear convergence for nonsmoothproblems. Second, we identify additional regularity, which we call the secondorder supporting hyperplane property (SOSH), that gives multiple-term quadraticconvergence. Third, we show that these fast convergence results carry over forthe convex inequality problem. Fourth, we show that infeasibility can bedetected in finitely many operations. Lastly, we explain how we can use thedual active set QP algorithm of Goldfarb and Idnani to get useful iterates bysolving the QPs partially, overcoming the problem of solving large QPs in ouralgorithms.

Improved analysis of algorithms based on supporting halfspaces and quadratic programming for the convex intersection and feasibility problems