bv_cvxbook_extra_exercises

# E the optimal solution of the whole coupled problem

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Unformatted text preview: tallation cost, and E f (y, b) is the expected optimal network ﬂow cost m πj f (y, b(j ) ), E f (y, b) = j =1 where f is the function deﬁned in part 1. Is (42) a convex optimization problem? 15.6 Maximizing algebraic connectivity of a graph. Let G = (V, E ) be a weighted undirected graph with n = |V | nodes, m = |E | edges, and weights w1 , . . . , wm ∈ R+ on the edges. If edge k connects nodes i and j , then deﬁne ak ∈ Rn as (ak )i = 1, (ak )j = −1, with other entries zero. The weighted Laplacian (matrix) of the graph is deﬁned as m wk ak aT = A diag(w)AT , k L= k=1 where A = [a1 · · · am ] ∈ Rn×m is the incidence matrix of the graph. Nonnegativity of the weights implies L 0. Denote the eigenvalues of the Laplacian L as λ1 ≤ λ2 ≤ · · · ≤ λn , which are functions of w. The minimum eigenvalue λ1 is always zero, while the second smallest eigenvalue λ2 is called the algebraic connectivity of G and is a measure of the connectedness of a graph: The larger λ2 is, the better connected the graph is. It is often used, for example, in analyzing the robustn...
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## This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.

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