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I want to solve \begin{align} \max |x| \text{ s.t } x^2 -4x + 4 = 0 \end{align}

Exercise 1 is to reformulate the problem without the norm. This yields \begin{align*} \max t \\ \text{s.t. } t-x &\geq 0\\ t+x &\geq 0 \\ x^2 -4x + 4 &= 0 \end{align*}

Exercise 2 is to use KKT-conditions to solve the problem. In my script those are only given for a minimization problem, hence I reformulate the problem to \begin{align*} \min -t \\ \text{s.t. } t-x &\geq 0\\ t+x &\geq 0 \\ x^2 -4x + 4 &= 0 \end{align*}

Then the Lagrangian is given by $L(t,x,\lambda) = -t -\lambda_1 (t-x) -\lambda_2 (t+x) -\lambda_3 (x^2 -2x + 4)$. But the derivative with respecto to $t$ leads $-1-\lambda_1 -\lambda_2 = 0$. Since $\lambda_1, \lambda_2 \geq 0$ this can never be satisfied. However the problem clearly has a solution namely $x = 2$. How can this be?

EDIT: As it was pointed out in the comments, the reformulation was wrong. Indeed, the variable $t$ is not bounded. I now tried to reformulate it as \begin{align*} \min t\\ \text{ s.t. } t-x &\geq 0 \\ t+x & \geq 0 \\ x^2 -4x + 4 &= 0 \end{align*}

Would this be correct? Would this generalize?

samabu
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1 Answers1

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Your reformulation is wrong. Indeed, $t$ can be chosen as big as possible for any feasible $x$. Speaking of feasible $x$, one could also note that the equation $x^2 - 2x + 4 = 0$ is not solvable (in the reals).

gerw
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  • Sorry, there was a typo in the constraint. It is supposed to be $x² -4x + 4$. – samabu Nov 13 '24 at 07:50
  • Could you please elaborate how the reformulation is wrong? – samabu Nov 13 '24 at 07:50
  • This is already in my answer. In your reformulation, you only have lower bounds on $t$, thus $t$ can be made as big as possible. Thus, the reformulated problem does not have a solution. – gerw Nov 13 '24 at 08:02
  • Ah I see. But isnt it the same approach as used here: https://math.stackexchange.com/questions/432003/converting-absolute-value-program-into-linear-program – samabu Nov 13 '24 at 08:07
  • How would you do this for a maximization problem? – samabu Nov 13 '24 at 08:22