Strictly Quasiconvex Function



Let f:SRn and S be a non-empty convex set in Rn then f is said to be strictly quasicovex function if for each x1,x2S with f(x1)f(x2), we have $f\left ( \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:SRn be strictly quasiconvex function and S be a non-empty convex set in Rn.Consider the problem: minf(x),xS. If ˆx is local optimal solution, then ˉx is global optimal solution.

Proof

Let there exists ˉxS such that f(ˉx)f(ˆx)

Since ˉx,ˆxS and S is convex set, therefore,

λˉx+(1λ)ˆxS,λ(0,1)

Since ˆx is local minima, f(ˆx)f(λˉx+(1λ)ˆx),λ(0,δ)

Since f is strictly quasiconvex.

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

Hence, it is contradiction.

Strictly quasiconcave function

Let f:SRn and S be a non-empty convex set in Rn, then f is saud to be strictly quasicovex function if for each x1,x2S with f(x1)f(x2), we have

f(λx1+(1λ)x2)>min{f(x1),f(x2)}

.

Examples

  • f(x)=x22

    It is a strictly quasiconvex function because if we take any two points x1,x2 in the domain that satisfy the constraints in the definition $f\left (\lambda x_1+\left (1- \lambda\right )x_2\right )

  • f(x)=x2

    It is not a strictly quasiconvex function because if we take take x1=1 and x2=1 and λ=0.5, then f(x1)=1=f(x2) but f(λx1+(1λ)x2)=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 f(λx1+(1λ)x2)>min{f(x1),f(x2)}. As the function is increasing in the negative x-axis and it is decreasing in the positive x-axis.

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