bv_cvxbook_extra_exercises

each edge can support power ow in either direction

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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 max-cut 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 max-cut problem is y T L(W )y = i≤ j wij (yi − yj )2 . If y is ±1-vector 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 max-cut 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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