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Unformatted text preview: onsider a network with n directed arcs. The flow through arc k is denoted xk and can be positive, negative, or zero. The flow vector x must satisfy the network constraint Ax = b where A is the node-arc incidence matrix and b is the external flow supplied to the nodes. Each arc has a positive capacity or width yk . The quantity |xk |/yk is the flow density in arc k . The cost of the flow in arc k depends on the flow density and the width of the arc, and is given by yk φk (|xk |/yk ), where φk is convex and nondecreasing on R+ . (a) Define f (y, b) as the optimal value of the network flow optimization problem n minimize k=1 subject to yk φk (|xk |/yk ) Ax = b with variable x, for given values of the arc widths y ≻ 0 and external flows b. Is f a convex function (jointly in y , b)? Carefully explain your answer. (b) Suppose b is a discrete random vector with possible values b(1) , . . . , b(m) . The probability that b = b(j ) is πj . Consider the problem of sizing the network (selecting the arc widths yk ) so that 132 the expected cost is minimized: minimize g (y ) + E f (y, b). (42) The variable is y . Here g is a convex function, representing the ins...
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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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