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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 optimization 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 CauchySchwarz 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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This note was uploaded on 09/03/2011 for the course EE 227A taught by Professor Staff during the Spring '11 term at University of California, Berkeley.
 Spring '11
 Staff
 Electrical Engineering

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