

T1. Surveillance Technologies Beyond StateoftheArt T2. Robust Resistor Network Topology Design by Conic Optimization
DETAILS T1. Surveillance Technologies Beyond StateoftheArt Topics:
T2. Robust Resistor Network Topology Design by Conic Optimization A resistance network is an electrical network comprised of resistors only. The resistors are linked to each other; the points at which they are linked to each other are called the nodes of the network. We consider the situation where the in and output currents at the nodes are given. Given the network and the external current we can compute the dissipation in the network, i.e., the energy consumption of the network. The dissipation highly depends on the topology of the network and the values of the resistors in the network, of course. Given the external current, our aim is to find a topology of the network and values of its resistors so that the dissipation is minimal. We call this the Resistance Network Topology Design Problem (RNTD) problem. As one may easily verify, a straightforward approach to the RTND problem yields a mathematical optimization model which is a nonconvex quadratic model. This makes the model in general hard to solve. The first surprise is that the RNTD problems can be modeled as a linear optimization problem, which makes it easy to solve. Let us call a resistor network optimal if it solves the RNTD problem, for some given external current f. One may wonder how the network behaves when it is fed with a current that differs slightly from the current f for which it is designed. We will demonstrate that very small perturbations in the design current may have a disastrous effect on the dissipation. We give a small example in which a perturbation of the design current of only 10% may lead to an increase of the dissipation by as much as a factor 21. The example suggests that the sensitivity of a network to perturbations in the current vector highly depends on the smallest eigenvalue of the socalled conductance matrix of the network. A possible approach to obtain more robust networks, i.e., networks that are less sensitive to perturbations in the external current, is to design a network that minimizes the maximal dissipation for a finite set of external currents. This is the multicurrent RNTD problem. When we choose the external currents close to the design current we might expect that the resulting network becomes more robust. It will be shown that the multicurrent RNTD problem can be modeled as a semidefinite optimization problem. For such problems there exist nowadays efficient (polynomialtime) solvers. A more natural approach to the robust RNTD problem is to allow the external current to belong to a small ellipsoid around the design current, and then to minimize the maximal dissipation when the external current runs through this ellipsoid. We conclude by presenting a semidefinite model for this problem, and some convincing examples. Content contributors: C. Roos; Faculty of Information Technology and Systems, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands (c.roos@ewi.tudelft.nl). Y. Bai: Shanghai University, Shanghai, China. Supported by Shanghai Leading Academic Discipline Project, No S30104 (yanqin.bai@gmail.com) D. Chaerani; Jurusan Matematika FMIPA Universitas Padjadjaran, Bandung, Indonesia (dchaerani@yahoo.com) 

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