
- Home
- Introduction
- Linear Programming
- Norm
- Inner Product
- Minima and Maxima
- Convex Set
- Affine Set
- Convex Hull
- Caratheodory Theorem
- Weierstrass Theorem
- Closest Point Theorem
- Fundamental Separation Theorem
- Convex Cones
- Polar Cone
- Conic Combination
- Polyhedral Set
- Extreme point of a convex set
- Direction
- Convex & Concave Function
- Jensen's Inequality
- Differentiable Convex Function
- Sufficient & Necessary Conditions for Global Optima
- Quasiconvex & Quasiconcave functions
- Differentiable Quasiconvex Function
- Strictly Quasiconvex Function
- Strongly Quasiconvex Function
- Pseudoconvex Function
- Convex Programming Problem
- Fritz-John Conditions
- Karush-Kuhn-Tucker Optimality Necessary Conditions
- Algorithms for Convex Problems
Strictly Quasiconvex Function
Let f:S→Rn and S be a non-empty convex set in Rn then f is said to be strictly quasicovex function if for each x1,x2∈S 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:S→Rn be strictly quasiconvex function and S be a non-empty convex set in Rn.Consider the problem: minf(x),x∈S. If ˆx is local optimal solution, then ˉx is global optimal solution.
Proof
Let there exists ˉx∈S such that f(ˉx)≤f(ˆx)
Since ˉx,ˆx∈S and S is convex set, therefore,
λˉx+(1−λ)ˆx∈S,∀λ∈(0,1)
Since ˆx is local minima, f(ˆx)≤f(λˉx+(1−λ)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:S→Rn and S be a non-empty convex set in Rn, then f is saud to be strictly quasicovex function if for each x1,x2∈S with f(x1)≠f(x2), we have
f(λx1+(1−λ)x2)>min{f(x1),f(x2)}
Examples
-
f(x)=x2−2
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+(1−λ)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+(1−λ)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.