(a) For an arbitrary point y = R", II(y) be the projection of y onto S. Find an expression for II(y) and give a short argument (i.e., proof) for why this is the correct expression. Make sure to handle the case y = 0 (i.e., the zero vector). Hint: Recall that the projection minimizes || -y|| for y € S. One approach would be to consider the reverse triangle inequality ||x − y|| ≥ |||r|| - || y||| and find a projection formula that achieves the global lower bound (i.e. equality instead of inequality). This would prove you've found the projection. (b) Is S a convex set? (c) Write a projected gradient descent algorithm, with constant step size μ, for ||||| = 1. min x¹Qx subject to TER" (d) Is the projected gradient descent algorithm guaranteed to converge to the solution for small enough ? If not, can you give an example of Q and an initialization (0) where the algorithm won't converge? Hint: Consider a diagonal matrix Q where not all entries are equal.

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