Solve optimization problem with constant cost

Question

Given that

$x_1 + 2x_2 + x_3 + 2x_4 =C$

with constraints

$x_1+2x_2\leq C_1$

$x_3+2x_4\leq C_2$

Find $x_1$, $x_2$, $x_3$ and $x_4$

My doubt is that can we have a solution for this type of problems?

Answer 1

A $4$-tuple $(x_1,x_2,x_3,x_4)$ satisfying the conditions exists if and only if $C\le C_1+C_2$.

Assuming $C\le C_1+C_2$, and assuming no further constraints, the feasible region is infinite.

Explicitly, assume $C\le C_1+C_2$, and let $w=(C_1+C_2)-C$.

Let $x_2,x_4$ be arbitrary real numbers, and let \begin{align*} x_1=C_1-2x_2-{\small{\frac{w}{2}}}\\[4pt] x_3=C_2-2x_4-{\small{\frac{w}{2}}}\\[4pt] \end{align*} Then all the specified conditions are satisfied.

On the other hand if $C > C_1+C_2$, summing the inequality constraints yields $$x_1+2x_2+x_3+2x_4 \le C_1+C_2 < C$$ contradicting the equality constraint.