The following is the mathematical expression for my model's rate expression. Variables $x,y$ are the controlling parameter, while the rest are positive constants.
$$\max_{x,y} \ ax + by^3 \ $$ $$\text{subject to}: \ 0\leq x \leq 1,\ 0\leq y\leq1$$
Can I mathematically say that it is a convex problem within the limits of variables $x,y$?
The graph for the equation strictly follows the definition of convexity. I have learned to solve the problems with KKT method, but I cannot understand how to resolve the inequality constraints $0 \leq (x,y) \leq 1$.
This is not a convex optimization problem. Actually it is a concave optimization (programming) problem.
The objective function, $ax + by^3$, is a convex function over the constraint region $ 0 \le x \le 1, 0 \le y \le 1$, but it is being maximized, which is equivalent to minimizing its negative, which is a concave function.
A concave function being minimized over compact convex constraints, as in this problem, has a global optimum at an extreme of the constraints. In this case, that means at $x = 0$ or $1$, and $y =0$ or $1$. Because $a$ and $b$ are both positive, the optimum occurs at $x = y = 1$, and has optimal objective value equal to $1$.