Question

4) Consider the following constrained optimization problem: S.1 2x, + 5x, + x, =3 a) (10pts) Determine a null space matrix Z for A (given the constraint set A b b) (25pts) Perform two iterations of the reduced Newton method starting from the point r0

Answer 1

(a) Null space matrix Z for A given the set Ax = b) Any solution Ax = b (if it)can be written as x = x_0 + z where z summationsign N(A) and no& N(o)^+ and x_0 is that is solution without nullspace Since x_0 + z, the phythagorem theorem gives that parallelto x parallelto ^2 = parallelto x_0 parallelto^2 + parallelto z parallelto^2 Here A = [2 5 1]^T x = [x_1 x_2 x_3] Ax = 0 rightdoublearrow 2x_1 + 5x_2 + x_3 = 0 rightdoublearrow x_3 = -2x_1 - 5x_2 Now, let x_1 = a, x_2 = b therefore solution is [a b -2a - 5b] = a[1 0 -2] + b[0 1 -5] So, Z = [1 0 -2 0 1 -5] is the Null space matrix \r\n Two of the reduded Newton method starting from the point x^0 = [0 0 3] satisfy the constrainly nabla f(x) = [2x_1 + 2pi_1 x^2_3 + 2x_2 2x_1 + 4x63_2 + 8 s