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(a) For an arbitrary point y E 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 E S. One approach would be to consider the reverse triangle inequality | - y|| ≥ |||*|| - || 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.

(a) For an arbitrary point y E 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 E S. One approach would be to consider the reverse triangle inequality | - y|| ≥ |||*|| - || 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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2. Consider the hollow sphere S in R", i.e., the set S = {x € R ||*||² = 1}. Consider the function f: RR given by f(x) = x¹Qx where Q is an n x n symmetric matrix. For this problem you may use the fact that Vf(x) = 2Qx. (a) For an arbitrary point y E 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 E S. One approach would be to consider the reverse triangle inequality ||x − y|| ≥ |||x|| - ||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 ||||² = 1. min x¹ Qx subject to ZERn (d) Is the projected gradient descent algorithm guaranteed to converge to the solution for small enough u? 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.
Transcribed Image Text:2. Consider the hollow sphere S in R", i.e., the set S = {x € R ||*||² = 1}. Consider the function f: RR given by f(x) = x¹Qx where Q is an n x n symmetric matrix. For this problem you may use the fact that Vf(x) = 2Qx. (a) For an arbitrary point y E 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 E S. One approach would be to consider the reverse triangle inequality ||x − y|| ≥ |||x|| - ||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 ||||² = 1. min x¹ Qx subject to ZERn (d) Is the projected gradient descent algorithm guaranteed to converge to the solution for small enough u? 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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