Establishing quasiconcavity

Question

Let $f:\mathbb{R}_+\rightarrow\mathbb{R}_+$ be twice differentiable quasi-concave function satisfying $f(x)>0,\forall x \in \mathbb{R}_+$. Let $g:\mathbb{R}_+\rightarrow\mathbb{R}_+$ be a positive, increasing and convex function (i.e. $g(x)>0,g'(x)>0,g''(x)>0$) satisfying $$ g(x)\leq f(x), \forall x\in [a,b] \text{ with, } a<b\in \mathbb{R}_+.$$ Is the sum $$ h(x)=f(x)-g(x)$$ quasiconcave on the interval $[a,b]$? Can you recommend any references beyond standard books on convex optimization?

Answer 1

The answer is negative. Take $[a,b]=[1,3]$. Let $f(x)=x+\epsilon(x-2)^2,$ where $\epsilon$ is small enough, so that the function is increasing, thus quasiconcave. Then take $g(x)=x+2\epsilon(x-2)^2-c$, where $c$ is such that $g\leq f$ on $[a,b]$. Then the difference $f(x)-g(x)=-\epsilon(x-2)^2+c$ has a strict maximum at $x=2$ thus it is not quasiconcave.

Answer 2

I will provide a criterion under which certain sums preserve quasiconcavity. Since it is not covered in traditional convex optimization textbooks ala Boyd & Vandenberghe (2004), since it is quite recent it might help others.

The reference which helped me out is: Quah, John K. H., and Bruno Strulovici. “AGGREGATING THE SINGLE CROSSING PROPERTY.” Econometrica, vol. 80, no. 5, 2012, pp. 2333–48. JSTOR, http://www.jstor.org/stable/23271450.

The idea is the following. A function $h(x)=f(x)+g(x), x\in X$ is quasiconcave when $h'(x)$ has the single-crossing (SC) property, i.e., it crosses the $x$-intercept only once. Theorem 1 of the above reference, is:

Let $f$ and $g$ be two SC functions. Then $\alpha f + \beta g$ is an (SC) function for any nonnegative scalars $\alpha$ and $\beta$ if and only if $f$ and $g$ obey signed-ratio monotonicity of the form:

(a) at any $x′\in X$ , such that $g(x′) < 0$ and $f(x′)>0$, we have $$\frac{−g(x′)}{f(x′)}\geq \frac{−g(x′′)}{f(x′′)}\text{ when } x′′ > x′$$ and (b) at any $x′\in X$ , such that $f(x′) < 0$ and $g(x′)>0$, we have $$\frac{−f(x′)}{g(x′)}\geq \frac{−f(x′′)}{f(x′′)}\text{ when } x′′ > x′$$