Projection methods for conic feasibility problems: applications to polynomial sum-of-squares decompositions

This paper presents a projection-based approach for solving conic feasibility problems. To find a point in the intersection of a cone and an affine subspace, we simply project a point onto this intersection. This projection is computed by dual algorithms operating a sequence of projections onto the cone, and generalizing the alternating projection method. We release an easy-to-use Matlab package implementing an elementary dual projection algorithm. Numerical experiments show that, for solving some semidefinite feasibility problems, the package is competitive with sophisticated conic programming software. We also provide a particular treatment of semidefinite feasibility problems modeling polynomial sum-of-squares decomposition problems.