Exercise 2: Review: Analytical Optimization
Date: Summer Term 2006
Unit 1: Optimize, i.e., locate the optima (that is maxima and minima), of the following three one-dimensional functions:
f(x) = 5x2−2x+ 10
g(x) = (x2−8x+ 16)(x−1) h(x) = sin(x)
What is the difficulty withh(x)?
Unit 2: The figure presented below depicts an electrical power supply network consisting of a transformer (T), four houses (H1..4), and three distribution nodes (D1..3). The positions, i.e., the xand y coordinates, of the transformer and the houses are fixed and can be found in the figure. The positions of the distribution nodes, however, are flexible and subject to minimization.
Transformer (3, 5)
House 4 (24, 20.6)
House 3 (22, 25.5) House 2 (16.2, 4.1)
Distributer 1
House 1 (10, 16.2)
x y
Distributer 2 Distributer 3
According to Pythagoras, the distancelpq between two points p and q is given as:
lpq = q(px−qx)2+ (py−qy)2. The total length of the network is the sum of the seven pieces:kT−D1k,kD1−H1k,kD1−D2k,kD2−H2k,kD2−D3k,kD3−H3k, andkD3−H4k.
Please, write down the formula for the total length L(xD1, yD1, xD2, yD2, xD3, yD3) of the entire network.
Questions: Can you solve the equation for the sixxi? What is the problem?
Have fun, Hagen and Ralf.
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