Convex function on R^n

Definition

A set $C \subseteq \mathbb{R}^n$ is convex if $t x + (1 - t) y \in C$ for all $x, y \in C$ and all $t \in [0, 1]$ . Let $C \subseteq \mathbb{R}^n$ be convex. A function $f \colon C \to \mathbb{R}$ is convex if, for all $x, y \in C$ and all $t \in [0, 1]$ , $f(t x + (1 - t) y) \le t f(x) + (1 - t) f(y).$ The function $f$ is strictly convex if the inequality is strict for all $x \ne y$ and all $t \in (0, 1)$ .

Remarks

If $C$ is open and $f$ is differentiable on $C$ , then $f$ is convex if and only if $f(y) \ge f(x) + \nabla f(x) \cdot (y - x)$ for all $x, y \in C$ . Let $C$ be open and let $f$ be of class $C^2$ . Then $f$ is convex if and only if its Hessian matrix is positive semidefinite at each point of $C$ . Each local minimum of a convex function is a global minimum, and a strictly convex function has at most one global minimum. A function can be convex in each variable separately but not convex. The function $f(x, y) = x y$ is linear in each variable, but $f(t, -t) = -t^2$ is not convex.

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Tag	`0M1GJSWZDA`
URI	`m2://0M1GJSWZDA`
Kind	definition
Writing	W0 stub
MSC	26B25
Defines	Convex function on R^n
Uses	Function of several variables

Convex function on R^n

Definition

Remarks

Reading path

Further reading

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