Numerical Methods and Optimization

176 Numerical Methods and Optimization: An Introduction

(0, +∞) such that x = y and any α ∈ (0, 1), the following inequality holds:

1

αx +(1− α)y

<

α

x

+

1 − α

y

∀α ∈ (0, 1),x,y ∈ (0, +∞),x = y. (8.15)

Multiplying both sides by (αx +(1− α)y)xy, which is positive, we obtain an

equivalent inequality

xy < (αx +(1− α)y)(αy +(1− α)x).

Since (αx+(1−α)y)(αy+(1−α)x)−xy = α(1−α)(x−y)

2

> 0, inequality (8.15)

is correct, hence f is strictly convex by definition.

Given X ⊂ IR

n

, discontinuities for a convex function f : X → IR c a n o n l y

occur at the boundary of X, as implied by the following theorem.

Theorem 8.3 A convex function is continuous in the interior int (X