Help needed with an optimisation problem using Kuhn-Tucker-Karush

Question

Maximise

$ y_1 + y_2 + y_3 $ subject to

$ y_1 +2y_3 \leq 1 $

$2y_1 + y_2 \leq 1 $

$2y_2 + y_3 \leq 1 $

$ y_1 \geq 0 , y_2 \geq 0, y_3 \geq 0 $

How would I do this with three constraints?

Answer 1

Guide:

• Sum up the three inequalities, you should get an upper bound.

• Prove that the upper bound is attainable by symmetry.

Remark:

If you insist to solve this using KKT, sure, there is almost no different from $2$ constraints, you just have more langrange multiplier.

Edit:

Dual problem is

$\min p_1 + p_2 + p_3$

subject to

$p_1 + 2p_2 \ge 1$

$p_2 + 2p_3 \ge 1$

$2p_1 + p_3 \ge 1$

$p_1 \ge 0, p_2 \ge 0, p_3 \ge 0$