* Exam 2008 problem #1 part (c) (to be done without utilizing any results from (a), (b))
* Exam 2009 problem #1 (a), (b)

From the compendium: 
* 1-07 (recall the definition of orthogonality: dot product = 0)
* 1-13
* 2-04


* Let g(y) = f(Ay), where y is an n-vector and A is a matrix of order m by n.
(a) Calculate the gradient and Hessian of g and express them as matrix products.  Be careful to get everything in the right order.
(b) Let (r_i transposed) be row number i of A, and let c_i be column number i.  Which one of f(r_i) or f(c_i) is well-defined for every m? Denote the well-defined one as z_i.
(c) Let h(y) = g(y) - (z transposed) y.  Calculate the gradient and Hessian of h.
(d) Suppose g is convex and that we are given the problem to maximize h subject to y being in a given compact (= closed and bounded) set C. Show that the maximum must be on the boundary.  
(e) use part (d) to find the maximum in the case where n = 2, and S is the set of all y_i ¡Ý 0, such that their sum is ¡Ü1.  
(In case of character set issues: the ¡Ý symbol is supposed to be greater-than-or-equal.) 



* Application case:

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A risk manager of a savings bank attempts to explain the potential for losses on mortgages in a potentially averse market:
	?In our mortgage portfolio, the average loan-to-value ratio is 80 percent, so even if our customers were totally broke, apart from the house pledged as collateral -- then we could stand a 20 percent price drop without any other losses than the administrative cost of collecting the collateral.?


Q: What is wrong with this argument? Formalise using Jensen's inequality.

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