Approximate the minimizer of the Rosenbrock function defined in Example 7.3 using the descent method

Approximate the minimizer of the Rosenbrock function defined
in Example 7.3 using the descent method with different choices of the descent
directions (7.35)–(7.38). Set x(0) = (_1.2, 1) and _
= 10_8 as tolerance for the stopping test, plot the
convergence histories for the different choices of the descent directions and
comment on the efficiency of the different methods.

Example 7.3 (The Rosenbrock function) The Rosenbrock
function  whose contour lines are displayed in Figure
7.7 ([Ros61]), is often used to test both efficiency and robustness of
minimization algorithms. Its global
»

Approximate the minimizer of the Rosenbrock function defined
in Example 7.3 using the descent method with different choices of the descent
directions (7.35)–(7.38). Set x(0) = (_1.2, 1) and _
= 10_8 as tolerance for the stopping test, plot the
convergence histories for the different choices of the descent directions and
comment on the efficiency of the different methods.

Example 7.3 (The Rosenbrock function) The Rosenbrock
function  whose contour lines are displayed in Figure
7.7 ([Ros61]), is often used to test both efficiency and robustness of
minimization algorithms. Its global minimum is attained at the point x_ = (1, 1),
however its variation around x_
is fairly low, making algorithms’ convergence quite problematic. Through the
following command

we would obtain additional information on the minimum value
of f  on the number of iterations,  as well
as the total number of function evaluations  Finally, the error
tolerance can be modified by using the command  as already discussed in
Example 7.2.

»

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