what is the difference between pseudo convexity and quasi convexity?

Question

I have difficulties in understanding the differences between pseudo convex and quasi convex functions. For example a fractional convex/concave function is pseudo convex or quasi convex?

Answer 1

For simplicity, let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$.

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$f$ is said to be convex if $$ f(\lambda x+\left(1-\lambda\right)y)\leq\lambda f(x)+\left(1-\lambda\right)f(y)\text{ for all }x,y\in\mathbb{R}^{n}\text{ and }\lambda\in[0,1]. $$ That is, for any points $x,y$, the line joining $(x,f(x))$ to $(y,f(y))$ has to lie above the function $f$.

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$f$ is said to be quasiconvex if $$ f(\lambda x+\left(1-\lambda\right)y)\leq\max\left\{ f(x),f(y)\right\} \text{ for all }x,y\in\mathbb{R}^{n}\text{ and }\lambda\in[0,1]. $$ That is, for any points $x,y$ such that (without loss of generality) $f(x)\geq f(y)$, the line joining $(x,f(x))$ to $(y,f(x))$ has to lie above the function $f$. Since the line joining $(x,f(x))$ to $(y,f(x))$ lies above the line joining $(x,f(x))$ to $(y,f(y))$, any convex function is trivially a quasiconvex function.

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$f$ is said to be pseudoconvex if it is once-differentiable and $$ \nabla f(x)^{\top}(y-x)\geq0\implies f(y)\geq f(x)\text{ for all }x,y\in\mathbb{R}^{n}. $$ Note that if $f$ is convex and once-differentiable, by the first-order characterization of convexity, $$ f(y)\geq f(x)+\nabla f(x)^{\top}(y-x)\text{ for all }x,y\in\mathbb{R}^{n}. $$ Therefore, $f$ is pseudoconvex. Note, however, that the converse is not necessarily true (consider $x\mapsto x+x^{3}$).

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It's also possible to show that a pseudoconvex function is strictly quasiconvex [1], which is a stronger requirement than being quasiconvex. A concise summary of the above facts is (for once-differentiable functions)

$$ \text{convex}\implies\text{pseudoconvex}\implies\text{strictly quasiconvex}\implies\text{quasiconvex} $$

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An expository paper that is a great starting point for learning about these functions is given below.

[1] Mangasarian, Olvi L. "Pseudo-convex functions." Journal of the Society for Industrial and Applied Mathematics, Series A: Control 3.2 (1965): 281-290.

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Addendum: It seems as though fractional convex functions are related to quasiconvexity and pseudoconvexity.