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

Theorem

A quasiconvex function $f:Srightarrow mathbb{R}^n$ on a non-empty convex set S in $mathbb{R}^n$ is strongly quasiconvex function if it is not constant on a line segment joining any points of S.

Proof

Let f is quasiconvex function and it is not constant on a line segment joining any points of S.

Suppose f is not strongly quasiconvex function.

There exist $x_1,x_2 in S$ with $x_1 neq x_2$ such that

$$fleft ( z right )geq maxleft { fleft ( x_1 right ), fleft ( x_2 right ) right }, forall z= lambda x_1+left ( 1-lambda right )x_2, lambda in left ( 0,1 right )$$

$Rightarrow fleft ( x_1 right )leq fleft ( z right )$ and $fleft ( x_2 right )leq fleft ( z right )$

Since f is not constant in $left [ x_1,z right ]$ and $left [z,x_2 right ] $

So there exists $u in left [ x_1,z right ]$ and $v=left [ z,x_2 right ]$

$$Rightarrow u= mu_1x_1+left ( 1-mu_1right )z,v=mu_2z+left ( 1- mu_2right )x_2$$

Since f is quasiconvex,

$$Rightarrow fleft ( u right )leq maxleft { fleft ( x_1 right ),f left ( z right ) right }=fleft ( z right ):: and ::f left ( v right ) leq max left { fleft ( z right ),fleft ( x_2 right ) right }$$

$$Rightarrow fleft ( u right )leq fleft ( z right ) :: and :: fleft ( v right )leq fleft ( z right )$$

$$Rightarrow max left { fleft ( u right ),fleft ( v right ) right } leq fleft ( z right )$$

But z is any point between u and v, if any of them are equal, then f is constant.

Therefore, $max left { fleft ( u right ),fleft ( v right ) right } leq fleft ( z right )$

which contradicts the quasiconvexity of f as $z in left [ u,v right ]$.

Hence f is strongly quasiconvex function.

Theorem

Let $f:Srightarrow mathbb{R}^n$ and S be a non-empty convex set in $mathbb{R}^n$. If $hat{x}$ is local optimal solution, then $hat{x}$ is unique global optimal solution.

Proof

Since a strong quasiconvex function is also strictly quasiconvex function, thus a local optimal solution is global optimal solution.

Uniqueness − Let f attains global optimal solution at two points $u,v in S$

$$Rightarrow fleft ( u right ) leq fleft ( x right ).forall x in S:: and ::fleft ( v right ) leq fleft ( x right ).forall x in S$$

If u is global optimal solution, $fleft ( u right )leq fleft ( v right )$ and $fleft ( v right )leq fleft ( uright )Rightarrow fleft ( u right )=fleft ( vright )$

$$fleft ( lambda u+left ( 1-lambdaright )vright )

which is a contradiction.

Hence there exists only one global optimal solution.

Remarks

• A strongly quasiconvex function is also strictly quasiconvex fucntion.
• A strictly convex function may or may not be strongly quasiconvex.
• A differentiable strictly convex is strongly quasiconvex.