Adding constraints in a constrained problem

Question

Consider a simplified version of a problem I am looking at: $$\min_{x, y, z, t_1, t_2, t_3} x - x^2 - y + y^2 - z + z^2 + t_1$$ subject to: $$ -x + x^2 \leq a + t_1$$ $$ -y + y^2 \leq b - t_2$$ $$ -z + z^2 \leq c - t_3$$ $$ t_1 = t_2 + t_3$$ $$ 0 \leq t_1, t_2, t_3 \leq d$$ where $a$ is known, $b + c = d$, but $b$ and $c$ are unknown. Given the setup, I am wondering if it is valid to reduce the constraints to: $$ -x + x^2 \leq a + t_1$$ $$ -y + y^2 -z + z^2 \leq d - t_1$$ $$ 0 \leq t_1 \leq d$$ so the exact values of $b$ and $c$ are not required and $t_2$ and $t_3$ can be ignored. Is this wishful thinking?

Answer 1

In general it is not a good idea to add inequalities.

Suppose $b=0.5, c=1, y=z=0.5$ and $t_2=t_3=1$.

The inequality $-y + y^2 -z + z^2 \leq d - t_1$ holds and

$ -y + y^2 \leq b - t_2$ doesn´t hold.