Exactness of sums of squares relaxations involving 3×3 matrices and Lorentz cones

AbstractWe consider a family Pm,n of cones of positive maps and a semidefinite relaxation of these cones. The cone Pm,n can be described as the set of those linear mappings from the space Rm into the space of real symmetric n×n matrices which map the m-dimensional Lorentz cone into the cone of real symmetric positive semidefinite matrices. We describe the cone Pm,n as a cone of nonnegative polynomials in several variables. We show that the considered semidefinite relaxation is in fact a sums of squares relaxation corresponding to this description of Pm,n. Our main result is that for n=3 the relaxation is exact. Hence it yields the exact result for optimisation problems over the cones Pm,3. In particular, the matrix ellipsoid problem for real symmetric 3×3 matrices can be rewritten as feasibility problem of a linear matrix inequality. For m⩾4,n⩾4 there exist points in Pm,n which do not lie in the semidefinite set corresponding to the relaxation. Hence the relaxation is exact if and only if min(n,m)⩽3