Strongly Quasiconvex Function

Let $f:S ightarrow mathbb{R}^n$ 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 $left ( x_1 ight ) eq left ( x_2 ight )$, we have $fleft ( lambda x_1+left ( 1-lambda ight )x_2 ight )< max :left { fleft ( x_1 ight ),fleft ( x_2 ight ) ight },forall lambda in left ( 0,1 ight )$

Theorem

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

Proof

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

Suppose f is not strongly quasiconvex function.

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

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

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

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

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

$$Rightarrow u= mu_1x_1+left ( 1-mu_1 ight )z,v=mu_2z+left ( 1- mu_2 ight )x_2$$

Since f is quasiconvex,

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

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

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

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

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

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

Hence f is strongly quasiconvex function.

Theorem

Let $f:S ightarrow mathbb{R}^n$ 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 $u,v in S$

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

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

$$fleft ( lambda u+left ( 1-lambda ight )v ight )< max left {fleft ( u ight ),fleft ( v ight ) ight }=fleft ( u ight )$$

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.