Pseudoconvex Function

Let $f:S ightarrow mathbb{R}$ be a differentiable function and S be a non-empty convex set in $mathbb{R}^n$, then f is said to be pseudoconvex if for each $x_1,x_2 in S$ with $igtriangledown fleft ( x_1 ight )^Tleft ( x_2-x_1 ight )geq 0$, we have $fleft ( x_2 ight )geq fleft ( x_1 ight )$, or equivalently if $fleft ( x_1 ight )>fleft ( x_2 ight )$ then $igtriangledown fleft ( x_1 ight )^Tleft ( x_2-x_1 ight )<0$

Pseudoconcave function

Let $f:S ightarrow mathbb{R}$ be a differentiable function and S be a non-empty convex set in $mathbb{R}^n$, then f is said to be pseudoconvex if for each $x_1, x_2 in S$ with $igtriangledown fleft ( x_1 ight )^Tleft ( x_2-x_1 ight )geq 0$, we have $fleft ( x_2 ight )leq fleft ( x_1 ight )$, or equivalently if $fleft ( x_1 ight )>fleft ( x_2 ight )$ then $igtriangledown fleft ( x_1 ight )^Tleft ( x_2-x_1 ight )>0$

Remarks

If a function is both pseudoconvex and pseudoconcave, then is is called pseudopnear.

A differentiable convex function is also pseudoconvex.

A pseudoconvex function may not be convex. For example,

$fleft ( x ight )=x+x^3$ is not convex. If $x_1 leq x_2,x_{1}^{3} leq x_{2}^{3}$

Thus,$igtriangledown fleft ( x_1 ight )^Tleft ( x_2-x_1 ight )=left ( 1+3x_{1}^{2} ight )left ( x_2-x_1 ight ) geq 0$

And, $fleft ( x_2 ight )-fleft ( x_1 ight )=left ( x_2-x_1 ight )+left ( x_{2}^{3} -x_{1}^{3} ight )geq 0$

$Rightarrow fleft ( x_2 ight )geq fleft ( x_1 ight )$

Thus, it is pseudoconvex.

A pseudoconvex function is strictly quasiconvex. Thus, every local minima of pseudoconvex is also global minima.

Strictly pseudoconvex function

Let $f:S ightarrow mathbb{R}$ be a differentiable function and S be a non-empty convex set in $mathbb{R}^n$, then f is said to be pseudoconvex if for each $x_1,x_2 in S$ with $igtriangledown fleft ( x_1 ight )^Tleft ( x_2-x_1 ight )geq 0$, we have $fleft ( x_2 ight )> fleft ( x_1 ight )$,or equivalently if $fleft ( x_1 ight )geq fleft ( x_2 ight )$ then $igtriangledown fleft ( x_1 ight )^Tleft ( x_2-x_1 ight )<0$

Theorem

Let f be a pseudoconvex function and suppose $igtriangledown fleft ( hat{x} ight )=0$ for some $hat{x} in S$, then $hat{x}$ is global optimal solution of f over S.

Proof

Let $hat{x}$ be a critical point of f, ie, $igtriangledown fleft ( hat{x} ight )=0$

Since f is pseudoconvex function, for $x in S,$ we have

$$igtriangledown fleft ( hat{x} ight )left ( x-hat{x} ight )=0 Rightarrow fleft ( hat{x} ight )leq fleft ( x ight ), forall x in S$$

Hence, $hat{x}$ is global optimal solution.

Remark

If f is strictly pseudoconvex function, $hat{x}$ is unique global optimal solution.

Theorem

If f is differentiable pseudoconvex function over S, then f is both strictly quasiconvex as well as quasiconvex function.

Remarks

The sum of two pseudoconvex fucntions defined on an open set S of $mathbb{R}^n$ may not be pseudoconvex.

Let $f:S ightarrow mathbb{R}$ be a quasiconvex function and S be a non-empty convex subset of $mathbb{R}^n$ then f is pseudoconvex if and only if every critical point is a global minima of f over S.

Let S be a non-empty convex subset of $mathbb{R}^n$ and $f:S ightarrow mathbb{R}$ be a function such that $igtriangledown fleft ( x ight ) eq 0$ for every $x in S$ then f is pseudoconvex if and only if it is a quasiconvex function.