Not sure if this is the right section for this but...
Minimize $\displaystyle f(x_1, x_2) = -x_1 + x_2$ subject to the following constraints,
$\displaystyle 0 \leq x_1 \leq a$, $\displaystyle 0 \leq x_2 \leq 1$ and $\displaystyle x_2 \geq x_1^2$.
Not sure if this is the right section for this but...
Minimize $\displaystyle f(x_1, x_2) = -x_1 + x_2$ subject to the following constraints,
$\displaystyle 0 \leq x_1 \leq a$, $\displaystyle 0 \leq x_2 \leq 1$ and $\displaystyle x_2 \geq x_1^2$.
To minimize this don't you want x2 as small as possible? If so, then x2=x1^2. Then you're minimizing x^2-x, which is at x=1/2. If your bounds don't permit this, don't you just need to check the boundary values and x=1/2?