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Convex Optimization by Lieven Vandenberghe, Stephen Boyd

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Chapter 2

Convex sets

2.1  Affine and convex sets

2.1.1  Lines and line segments

Suppose x1 x2 are two points in Rn. Points of the form

where θ R, form the line passing through x1 and x2. The parameter value θ = 0 corresponds to y = x2, and the parameter value θ = 1 corresponds to y = x1. Values of the parameter θ between 0 and 1 correspond to the (closed) line segment between x1 and x2.

Expressing y in the form

gives another interpretation: y is the sum of the base point x2 (corresponding to θ = 0) and the direction x1 x2 (which points from ...

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