# Minimization of Quasiconcave Function

I have a function given below:

$$f(x,y) = \log_2(1 - xe^y + c + ax + axe^y)$$

I was hoping to find $$(x,y)$$ that minimizes this function. The constraints are $$0 < x <1$$ and $$y > 0$$. Here, both $$x$$ and $$y$$ are discrete. I wasn't sure if the function is convex or not. So, I tried checking and I think the function is $$quasiconcave$$. How can I find $$(x,y)$$ in this case such that it minimizes $$f$$?

• whether the problem has a solution will depend on the value of $a$. Is $a<1$? Mar 4, 2020 at 16:48
• yes, $a < 1$. Does it have a solution? Mar 4, 2020 at 16:50
• Yes, it does if $x$ and $y$ are continuous. I am not sure, though, what values you are allowed to give to $x$ and $y$ since you mention that they take discrete values. Mar 4, 2020 at 16:52
• x can be between $0$ and $1$ and y can be greater than $0$ Mar 4, 2020 at 16:53
• Do you mean $x\in[0,1]$ and $y\geq 0$ (you have strict inequalities in the question). But, then, the problem is not discrete. Mar 4, 2020 at 16:54

Hint: Notice that $$\log_2$$ is a strictly increasing function. The same holds for $$x\mapsto x+c$$.
• Are you implying that the $(x,y)$ that minimizes $f$ is a corner solution? As $\log_2$ is a strictly increasing function, the solution would be the starting point? Mar 4, 2020 at 16:46
• Well... it should be the solution that minimizes the argument. How does the expression "inside" the $\log_2$ behave? Mar 4, 2020 at 16:48
• I think the expression inside the $\log_2$ is not strictly increasing. It increases and then decrease Mar 4, 2020 at 16:52
• As a function of $x$ or of $y$? Try to factor out what you can. Mar 4, 2020 at 16:53