Let $f:Srightarrow mathbb{R}$ 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 $bigtriangledown fleft ( x_1 right )^Tleft ( x_2-x_1 right )geq 0$, we have $fleft ( x_2 right )geq fleft ( x_1 right )$, or equivalently if $fleft ( x_1 right )>fleft ( x_2 right )$ then $bigtriangledown fleft ( x_1 right )^Tleft ( x_2-x_1 right )

Pseudoconcave function

Let $f:Srightarrow mathbb{R}$ 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 $bigtriangledown fleft ( x_1 right )^Tleft ( x_2-x_1 right )geq 0$, we have $fleft ( x_2 right )leq fleft ( x_1 right )$, or equivalently if $fleft ( x_1 right )>fleft ( x_2 right )$ then $bigtriangledown fleft ( x_1 right )^Tleft ( x_2-x_1 right )>0$

Remarks

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

• A differentiable convex function is also pseudoconvex.

• A pseudoconvex function may not be convex. For example,

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

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

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

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

  Thus, it is pseudoconvex.

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

Strictly pseudoconvex function

Let $f:Srightarrow mathbb{R}$ 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 $bigtriangledown fleft ( x_1 right )^Tleft ( x_2-x_1 right )geq 0$, we have $fleft ( x_2 right )> fleft ( x_1 right )$,or equivalently if $fleft ( x_1 right )geq fleft ( x_2 right )$ then $bigtriangledown fleft ( x_1 right )^Tleft ( x_2-x_1 right )

Theorem

Let f be a pseudoconvex function and suppose $bigtriangledown fleft ( hat{x}right )=0$ 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, $bigtriangledown fleft ( hat{x}right )=0$

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

$$bigtriangledown fleft ( hat{x}right )left ( x-hat{x}right )=0 Rightarrow fleft ( hat{x}right )leq fleft ( xright ), forall x in S$$

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 $f:Srightarrow mathbb{R}$ 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 $f:Srightarrow mathbb{R}$ be a function such that $bigtriangledown fleft ( xright )neq 0$ for every $x in S$ then f is pseudoconvex if and only if it is a quasiconvex function.