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An Introduction to Optimization, 4th Edition by Stanislaw H. Zak, Edwin K. P. Chong

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Chapter 4

CONCEPTS FROM GEOMETRY

4.1 Line Segments

In the following analysis we concern ourselves only with n. The elements of this space are the n-component vectors x = [x1, x2,…, xn].

The line segment between two points x and y in n is the set of points on the straight line joining points x and y (see Figure 4.1). Note that if z lies on the line segment between x and y, then

Figure 4.1 Line segment.

equation

where α is a real number from the interval [0,1]. The equation above can be rewritten as z = αx + (1 − α)y. Hence, the line segment between x and y can be represented as

equation

4.2 Hyperplanes and Linear Varieties

Let u1, u2,…, un, v , where at least one of the ui is nonzero. The set of all points ...

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