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hw5_ee227a_rev2

# hw5_ee227a_rev2 - Consider the problem min x ∈ R n f x...

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UC Berkeley Department of Electrical Engineering and Computer Science EECS 227A Nonlinear and Convex Optimization Problem Set 5 Fall 2009 Issued: Thursday, October 22 Due: Thursday, November 5, 2009 Reading: Boyd and Vandenberghe, Chapter 5 Problem 5.1 Use a Lagrange multiplier approach to solve the problem min x R n n summationdisplay i =1 x i such that h ( x ) = bardbl x bardbl 2 2 1 = 0. Give a geometric interpretation of the Lagrange multiplier conditions. Problem 5.2 Given an n × n symmetric matrix Q , define e 1 = arg min bardbl x bardbl 2 =1 x T Qx and λ 1 = min bardbl x bardbl 2 =1 x T Qx, and for k = 1 , 2 , . . . , n 1, e k +1 = arg min bardbl x bardbl 2 =1 x T Qx such that e T i x = 0, i = 1 , . . . , k , and λ k +1 = min bardbl x bardbl 2 =1 x T Qx such that e T i x = 0, i = 1 , . . . , k . Using optimization principles and theory: (a) Show that λ 1 λ 2 ≤ · · · ≤ λ n . (b) Show that the vectors e 1 , . . . , e n are linearly independent. (c) Show how λ 1 , . . . , λ n can be interpreted as Lagrange mulipliers. Problem 5.3 This exercise develops the fact for problems with only linear constraints, the existence of Lagrange multipliers is guaranteed even when a local minimum is not regular.

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Unformatted text preview: Consider the problem min x ∈ R n f ( x ) subject to Ax = b , where A ∈ R m × n and b ∈ R m . (a) Show that of rank( A ) = k < m , then no local minimum can be regular. (b) Show that whether or not x ∗ is a regular local minimum, there exists some λ ∗ ∈ R m such that ∇ f ( x ∗ ) + A T λ ∗ = 0. (c) Show that λ ∗ need not be unique. 1 Problem 5.4 Given a vector y ∈ R n , consider the optimization problem max x ∈ R n y T x subject to x T Qx ≤ 1, where Q ≻ 0 is a symmetric PD matrix. Show that the optimal value of this optimiza-tion problem is r y T Q − 1 y and conclude that ( y T x ) 2 ≤ ( x T Qx )( y T Q − 1 y ). ( Note: This is a generalization of the usual Cauchy-Schwarz inequality, which corresponds to the special case Q = I .) Problem 5.5 B & V, Problem 5.6 Problem 5.6 B & V, Problem 5.11 Problem 5.7 B & V, Problem 5.14 2...
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