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Unformatted text preview: W ) = inf nλmax (L(W ) + diag(x)) ,
1T x=0 is convex. 128 (b) Give a simple argument why f (W ) is an upper bound on the optimal value of the combinatorial
optimization problem
maximize y T L(W )y
subject to yi ∈ {−1, 1}, i = 1, . . . , n. This problem is known as the maxcut problem, for the following reason. Every vector y
with components ±1 can be interpreted as a partition of the nodes of the graph in a set
S = {i  yi = 1} and a set T = {i  yi = −1}. Such a partition is called a cut of the graph.
The objective function in the maxcut problem is
y T L(W )y =
i≤ j wij (yi − yj )2 . If y is ±1vector corresponding to a partition in sets S and T , then y T L(W )y equals four
times the sum of the weights of the edges that join a point in S to a point in T . This is called
the weight of the cut deﬁned by y . The solution of the maxcut problem is the cut with the
maximum weight.
(c) The function f deﬁned in part 1 can be evaluated, for a given W , by solving the optimiz...
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 Fall '13
 F.Borrelli
 The Aeneid

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