Convex Functions and Proximation, Projection and Separation Theorems

Abstract

Convexity plays a crucial role in the study of minimisation problems. After defining convex functions and describing their elementary properties, we show that continuous convex functions are locally Lipschitz (Lipschitz in a suitable neighbourhood of each point). We then prove the theorem for the existence and uniqueness of a solution of the minimisation problem $$ \frac{1}{2}{\left\| {\bar x - {x_0}} \right\|^2} + f\left( {\bar x} \right) = \mathop {\inf }\limits_{x \in X} \left( {\frac{1}{2}{{\left\| {x - {x_0}} \right\|}^2} + f\left( x \right)} \right)$$ when f is a nontrivial convex lower semi-continuous function from X to ℝ ∪ {+∞}.