Problem 2. Explain how the following problem can be solved using a QR factorization of the matrix A: minimize ||Ac - b)|2 + ||Ay - cl|2 subject to dix = ely. The m xn matrix A has linearly independent columns and the n-vectors d, e are not zero. The variables in the problem are the n-vectors r and y. Clearly state the different steps in the algorithm and their complexity, including terms that are cubic (m3, man, mn2, n3) or quadratic (m2, mn, n2) in the dimensions. Solution. The optimality conditions for this constrained least, squares problem are AT A 0 0 AT A d' el' From the first two equations I = (ATA) '(ATb - dz), y = (ATA) '(ATc + ez). Substituting this in the third equation allows us to solve for z: d''(AT A) 'ATb - eT(ATA) 'ATC Z d'T (AT A) 'd + er(AT A) le