Constrained optimization with alternates in special conditions

Question

I have the following optimization problem.

$$\max_{a b} acx+bdy+z \ \ \ \ \ \ $$

subjected to

$$ c = \begin{cases} 1, & \text{if } 2xa-yb-z\geq 4\\ 0, & \text{if} \ 2ax<yb+z\\ \frac{2xa-yb-z}{4},& \text{otherwise}\\ \end{cases} $$

$$ d = \begin{cases} 1, & \text{if } xc-yb-z\geq 2\\ 0, & \text{if} \ xc<yb+z\\ \frac{xc-yb-z}{2},& \text{otherwise}\\ \end{cases} $$

$$ 0\leq a \leq1 \ \ \ \ \ \ $$ $$ 0\leq b \leq1 \ \ \ \ \ \ $$

I'm using matlab to optimize $a$ and $b$ using fmincon function. I can define $a$ and $b$ as bound constraints but I'm confused where should I define the function $c$.

Thanks in advance for any help and/or clue.

Answer 1

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