# Maximization when parameters are unknown

I would like to know if my understanding about how to find a maximum of the function when some parameters are unknown is correct.

Consider the following maximization problem.

$$\max_{x}V=\int_0^{a(x)}y(x)dF(t)+\int_{a(x)}^{\infty}z(x)dF(t)$$

where $$x\geq0$$, $$y'(x)>0$$, and $$z'(x)>0.$$ If $$x\to\infty$$, $$z(x)\to \overline{z}$$. $$F$$ is the cumulative distribution function of $$t$$.

The function $$a(x)$$ is non-monotonic in $$x$$:

if $$x=0$$, $$a(x)=0$$,

if $$0, $$a(x)>0$$,

and if $$x\to\infty$$, $$a(x)=0$$.

In this maximization problem, if we increase $$x$$, the first term in the right side first increases and then decreases to zero because of the nonmonotonicity of $$a(x)$$.

If $$y(x^*)+z(x^*)> y(x) +z(x)$$ for any $$x$$, the objective function $$V$$ is maximized at $$x=x^*$$, and the maximum value is $$\int_0^{a(x^*)}y(x^*)dF(t)+\int_{a(x^*)}^{\infty}z(x^*)dF(t)$$.

If $$y(x)+z(x)<\overline{z}$$ for any $$x$$, the objective function $$V$$ is maximized at $$x=\infty$$, and the maximum value is $$\int_{0}^{\infty}\overline{z}dF(t)$$. The first term in the right side is zero.

I would appreciate it if someone could tell me if my understanding is correct. If it is incorrect, where did I make a mistake?

• There seems to be a problem with your variable $t$ of integration which does not appear as argument of $y$ and $z$. – Bertrand May 23 '19 at 18:54
• did you mean to say $dF(x)$? – Regio May 23 '19 at 19:00

Disclaimer: I'm assuming you are integrating over $$x$$:
I have a problem with your first inequality. Since $$y(x)$$ and $$z(x)$$ are monotonically increasing (strictly) with respect to $$x$$, there cannot exists an $$x^*$$ such that $$y(x^*)+z(x^*)>y(x)+z(x)$$ for all $$x\geq 0$$. So the statement is vacuously true.
Note as well that even if $$y(x)>\bar z$$ for some values of $$x$$, the solution can still be at $$x=\infty$$ (assuming you can choose such a value of $$x$$, of course).