# Topkis' Theorem

Suppose my optimization problem is stated as follows

$\max\limits_x f(x,t)$

$s.t.$ $g(x,t) \leq 0$

I am interested in finding the direction $x^*$ changes with the parameter $t$. Can someone provide me a reference that describes conditions that if satisfied by $f$ and $g$ will let me answer the question of monotonicity of $x^*$ with respect to $t$?

$$\frac{\partial ^2 f(x^*,t)}{\partial x\partial t} \cdot \frac {\partial x^*}{\partial t} > 0$$
Topkis's Theorem is a more abstract mathematical treatment that states the condition required in order to have $\partial x^*/\partial t \geq0$.