Definition 2.1: Convex Set

A set $X\subset {ℝ}^n$ is said to be convex if $\alpha x+(1-\alpha )y\in X\forall <!--FUNCTION\; APPLICATION-->x,y\in X\alpha \in [0,1]$

Definition 2.2: Convex Function

A function $f:X\subset {ℝ}^n\to ℝ$ is convex if $X$ is convex and holds that:

Remark 2.1

A function is said strictly convex if the convexity condition holds with strict inequality, that is :

Definition 2.3: Cone, Convex Cone

A nonempty set $C\subset {ℝ}^n$ is said to be a cone if, for every point $x\in C$ ,

If, in addition, $C$ is convex then it is said to be a convex cone.

Definition 2.4: Positive definite matrix

A symmetric square matrix $A\in {ℝ}^n$ is said to be positive definite if $x{}^{}\top <!--FUNCTION\; APPLICATION-->Ax>0\forall <!--FUNCTION\; APPLICATION-->x\in {ℝ}^n$

Remark 2.2

Equivalently, positive definiteness implies that all the eigenvalues are positive. In the same way we define semi-positive definite matrices. A symmetric square matrix $A\in {ℝ}^n$ is said to be positive definite if $x{}^{}\top <!--FUNCTION\; APPLICATION-->Ax\ge 0\forall <!--FUNCTION\; APPLICATION-->x\in {ℝ}^n$ . To denote positive and semi-positive definite matrices we use the symbol $A\succ 0$ and $A\succcurlyeq 0$ respectively. If a symmetric matrix has both positive and negative eigenvalues is said to be indefinite.

Definition 2.5: small o

For a function $f:ℝ\to ℝ$ we write $f\left(x\right)=o\left(g\right(x\left)\right)$ if and only if there exists a neighbourhood of $0$ (i.e. ${\mathcal{ℬ}}_{𝜖}\left(0\right)$ ) and a function $c:{\mathcal{ℬ}}_{𝜖}\left(0\right)\to ℝ$ with $\underset{x\to 0}{\lim <!--FUNCTION\; APPLICATION-->}c\left(x\right)=0$ such that:

Definition 2.6

Matrix norm