max v GFk(x+t(y x)) hy x,vidt

The primal and dual of Lasserre's hierarchy(MomOpt-d) nicely illustrate the duality between momentsandpositivepolynomials. Recall that the subsets we defined areI1 = {1,2}, I2 = {2,3}. Now suppose that we need to consider an additional "bad" constraint(1 x1 x2 x3)2 = 0. AsdiscussedattheendofSection4,for2-hiddenlayernetworks,oneneedstoreducetheobjective function to degree 2 so that theHR-2 algorithm can be adapted to problem(LCEP). The moments of cubic termstiuj1uk2 lie in the second-order moment matrixM2(y,{ti,uj1,uk2}), which is of size 3+22 = 10. A variant of this technique is to enlarge the size ofthemoment matrices butinthemeantime reducethenumber ofmoment matrices.