7. 10. 11. 12. 13A Newton's method on a convex function, under lower and upper bounds on the eigenvalues of the Hessian, and a Lipschitz Hessian, converges (with suitable step sizes) at the rate: E at 0(loglog(1/()); E b. 0(log log(1/)), locally; E cl it depends on whether the function is selfconcordant; E (1. none of the above A Let 1\" denote the optimal criterion value of the convex problem miln f(I) subject to hj(z) g 0, j: 1,.'.,m, and 12*(t) denote the solution in the barrier problem mm mm) + am. Then f(a:*(t)) f* g m/t. E True E False A Each main iteration of the barrier method performs just one one Newton update; E True E False The main idea behind the barrier method is add terms to the criterion that: E at smoothly approximate indicator functions of the constraints; E b, make the new criterion strongly convex; E Co make the new criterion smooth; E d. get rid of equality constraints. The barrier method solves the problem: min f(:) subject to hj(.z) g 0, j: 1,.4.,m, by solving a: E a. single problem of the form min1(tf(ar) + #1)), where (:r) = 7 Zilloyihz and t > 0; E b. sequence of problems of the form minI (tkf(x) + MID, where (15(33) : 2:11 log(hj(x)) and 13k a 00; E oi sequence of problems of the form min: (tkas) + 915W, Where (15(Z) : 7 2;":1 log(7hj(1:)) and 1'1: > 0; E d sequence of problems of the fonn mini (tkm) W 45(1)), where (3:) = 2:1 log(7hj(3:)) and t}; 4) 00; E e. sequence of problems of the form min; (tkm) W (r)), where 125(2) : 211:1 log(7hj(3:)) and i}; a 0' For constrained convex minimization, barrier methods approach the solution from the outside of the constraint set. E True E False Which of the following statements about the barrier method and the primal~dual interiorpoint method ' not true (for convex problems)? E a. Both barrier method and primal-dual interior-point method can be interpreted as solving a perturbed version of the KKT conditions. E b. Both methods have local O(log(1/e)) rate of convergence. E c. Primal-de interior~point method is more commonly used in practice because it tends to be more Hicienti d. Both methods perform just one Newton update before taking a step along the central path (adjusting the barrier parameter t)r ._. (D Il