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Question 3 (Recourse function, (20 marks)). Let A be some set of scenarios and X CR" be an arbitrary set. Consider the risk-neutral stochastic program min ce +EQE,.)] where the recourse function Q: X x Q R is given by Q(x,w) = min foly,w) s.t. 1.9,w') +91(,w) Sh(w), i = 1,...,m, where Y is a convex set and for every w and i = 1,...,m, 9(w) is a convex function of t. 1. Suppose that for every w EA and i = 0,...,m, L.) is a convex function of y. Prove that QW) is a convex function of for every w (6 marks) + 2. Suppose that for every w E A and 1 = 0,......(W) is a concave function of y. Also assume that Y is a polytope, which is a convex set with finitely many extreme points!. Let the extreme points of Y be the vectors {y'.....} for some finite integer K. Prove that for every > 0, the recourse function can be lower bounded by the function 4: XXRR which is defined as (14 marks) (, , ) :=-.. () + 9 (, ) + mini *.) + .J.*). Also argue that this lower bound is a convex function of r for every 4 >0 and w 12. An extreme point in a point that cannot be written as a convex combination of two other points in the set. ,w) + ). Question 3 (Recourse function, (20 marks)). Let A be some set of scenarios and X CR" be an arbitrary set. Consider the risk-neutral stochastic program min ce +EQE,.)] where the recourse function Q: X x Q R is given by Q(x,w) = min foly,w) s.t. 1.9,w') +91(,w) Sh(w), i = 1,...,m, where Y is a convex set and for every w and i = 1,...,m, 9(w) is a convex function of t. 1. Suppose that for every w EA and i = 0,...,m, L.) is a convex function of y. Prove that QW) is a convex function of for every w (6 marks) + 2. Suppose that for every w E A and 1 = 0,......(W) is a concave function of y. Also assume that Y is a polytope, which is a convex set with finitely many extreme points!. Let the extreme points of Y be the vectors {y'.....} for some finite integer K. Prove that for every > 0, the recourse function can be lower bounded by the function 4: XXRR which is defined as (14 marks) (, , ) :=-.. () + 9 (, ) + mini *.) + .J.*). Also argue that this lower bound is a convex function of r for every 4 >0 and w 12. An extreme point in a point that cannot be written as a convex combination of two other points in the set. ,w) + )

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