Convex Optimization 3 -- Convex functions

Outline

• basic properties and examples
• operations that preserve convexity
• the conjugate function
• quasiconvex functions
• log-concave and log-convex functions
• convexity with respect to generalized inequalities

1.Definition

f:\mathbb{R}^n\to\mathbb{R} is convex if \textbf{dom} f is a convex set and
f(\theta x+(1-\theta)y)\leq\theta f(x)+(1-\theta)f(y)
for all x,y\in\textbf{dom} f,0\leq\theta\leq 1