An Introduction to Optimization, 4th Edition

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,…, x_n].

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.

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

4.2 Hyperplanes and Linear Varieties

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