A Cutting-Plane Alternating Projections Algorithm A Cutting-Plane, Alternating Projections Algorithm for Conic Optimization Problems

We introduce a hybrid projection-localization method for solving large convex cone programs. The method interleaves a series of projections onto localization sets with the classical alternating projections method for convex feasibility problems; the problem is made amenable to the method by reducing it to a convex feasibility problem composed of a subspace and a cone via its homogeneous self-dual embedding. At each step, the only requirement on the localization set, besides convexity, is that it contain the intersection of the subspace and the cone that define the feasibility problem. The key task, then, is to instantiate localization sets that are both informative and easy to project on — there is a trade-off between these two desiderata. Our primary contributions are three-fold. First, we present the projection-localization algorithm family and prove that it is convergent. Second, we empirically evaluate the algorithm obtained when the localization sets are polyhedrons defined by cutting planes obtained from projecting upon the subspace and the convex cone. Our findings are promising, but not yet entirely satisfying: our method significantly outperforms alternating projections on complex problems, where the cone is a cartesian product of multiple cones, but either matches or underperforms the alternating directions method of multipliers. For very simple problems, we demonstrate empirically that this method is, at least in appearance, quadratically convergent. Finally, not to be overlooked, our third contribution is the publication of software that enables clients to rapidly prototype and evaluate projection algorithms for convex feasibility and conic problems.