Kuhn Tucker optimization problem
Here is the Lagrangian function for the stated optimization problem:
$$\mathcal{L}(x_i, t) = v(x_i, t) - cx_i + \mu x_i$$

Necessary conditions for optimality:

$$\frac{\partial\mathcal{L}}{\partial x_i} =\frac{\partial v}{\partial x_i} - c + \mu = 0 $$
and
$$x_i\geq 0, \ \mu \geq 0, \ \mu x_i = 0$$

To solve it, consider the following cases :


$x_i &gt; 0$

$x_i &gt; 0 \rightarrow \mu = 0 \rightarrow \dfrac{\partial v}{\partial x_i} - c = 0$

If there exists $x_i^* &gt; 0$ such that $\dfrac{\partial v}{\partial x_i}\Big\vert_{x_i=x_i^*} - c = 0$, then $x_i = x_i^*$ solves the problem.
$x_i = 0$

$x_i = 0 \rightarrow \mu = c - \dfrac{\partial v}{\partial x_i}$

If $c - \dfrac{\partial v}{\partial x_i}\Big\vert_{x_i=0} \geq 0$, then $x_i = 0$ solves the problem.