1) minimize the function
f(x1, x2) = 5x1^ 2 + 5x2^ 2 - x1x2 -11x1 + 11x2 + 11
(a) Find a point satisfying the first-order necessary conditions for a solution.
(b) What kind of minima is the point you found in part (a)?
(c) What would be the worst rate of convergence for the steepest
descent method for this
problem?
(d) Starting at (x1, x2) = (0, 0), at worst how many steepest descent
iterations would it take
to reduce the function value to 10^-11?


2)  minimize the function f(x) = 1/2x^tQx + q^tx, where Q is a symmetric
matrix that is positive definite. Now suppose further that we perform
a linear transformation
of variables x = Sy, where S is some invertible n × n matrix. 
(a) Write down the equivalent minimization problem in terms of the
variables y, and denote the function that you are minimizing by f(y)

(b) If the steepest descent algorithm is used to minimize f(y) what
type of convergence will
it have? What will the appropriate constant depend on? What would be
the ideal choice for
S and why?  (HINT: The following fact from linear
algebra should help: any symmetric matrix Q with real entries has n eigenvalues
corresponding to n distinct eigenvectors which form an orthonormal
basis of Rn i.e. if R is
the matrix of eigenvectors then R^tR=I or equivalently R^-1 =R^t. In
addition, Q =RDR^t where
D is a diagonal matrix of the corresponding eigenvalues of Q.)

(c) Write down the formula for the (pure i.e. step length is 1) Newton
iterate update for the
functions f(x) and f(y)

(d) Generalize the result in part (c) for Newton?s method for a
general function f(x) and f(y) derived by substituting x = Sy.