Let $f:Srightarrow mathbb{R}^n$ and S be a non-empty convex set in $mathbb{R}^n$ then f is said to be strictly quasicovex function if for each $x_1,x_2 in S$ with $fleft ( x_1 right ) neq fleft ( x_2 right )$, we have $fleft ( lambda x_1+left ( 1-lambda right )x_2 right )

Remarks

• Every strictly quasiconvex function is strictly convex.
• Strictly quasiconvex function does not imply quasiconvexity.
• Strictly quasiconvex function may not be strongly quasiconvex.
• Pseudoconvex function is a strictly quasiconvex function.

Theorem

Let $f:Srightarrow mathbb{R}^n$ be strictly quasiconvex function and S be a non-empty convex set in $mathbb{R}^n$.Consider the problem: $min :fleft ( x right ), x in S$. If $hat{x}$ is local optimal solution, then $bar{x}$ is global optimal solution.

Proof

Let there exists $ bar{x} in S$ such that $fleft ( bar{x}right )leq f left ( hat{x}right )$

Since $bar{x},hat{x} in S$ and S is convex set, therefore,

$$lambda bar{x}+left ( 1-lambda right )hat{x}in S, forall lambda in left ( 0,1 right )$$

Since $hat{x}$ is local minima, $fleft ( hat{x} right ) leq fleft ( lambda bar{x}+left ( 1-lambda right )hat{x} right ), forall lambda in left ( 0,delta right )$

Since f is strictly quasiconvex.

$$fleft ( lambda bar{x}+left ( 1-lambda right )hat{x} right )

Hence, it is contradiction.

Strictly quasiconcave function

Let $f:Srightarrow mathbb{R}^n$ and S be a non-empty convex set in $mathbb{R}^n$, then f is saud to be strictly quasicovex function if for each $x_1,x_2 in S$ with $fleft (x_1right )neq fleft (x_2right )$, we have

$$fleft (lambda x_1+left (1-lambdaright )x_2right )> min left { f left (x_1right ),fleft (x_2right )right }$$.

Examples

• $fleft (xright )=x^2-2$

  It is a strictly quasiconvex function because if we take any two points $x_1,x_2$ in the domain that satisfy the constraints in the definition $fleft (lambda x_1+left (1- lambdaright )x_2right )

• $fleft (xright )=-x^2$

  It is not a strictly quasiconvex function because if we take take $x_1=1$ and $x_2=-1$ and $lambda=0.5$, then $fleft (x_1right )=-1=fleft (x_2right )$ but $fleft (lambda x_1+left (1- lambdaright )x_2right )=0$ Therefore it does not satisfy the conditions stated in the definition. But it is a quasiconcave function because if we take any two points in the domain that satisfy the constraints in the definition $fleft ( lambda x_1+left (1-lambdaright )x_2right )> min left { f left (x_1right ),fleft (x_2right )right }$. As the function is increasing in the negative x-axis and it is decreasing in the positive x-axis.