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How can I evaluate the minimum of $$ \left|7x-1\right|+\left|7y-5\right|+\left|7z-1\right| $$ if $x,y,z$ are non negative reals such that $ x+y+z=1$ and $y^2 \le xz$?

Is there a standard way to solve such kind of optimization? I putted here random coefficients which do not satisfy the inequality on the constraint so that the answer shouldn't be trivially zero..

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  • $\begingroup$ Minimize[{Abs[7 x - 1] + Abs[7 y - 5] + Abs[7 z - 1], x + y + z == 1, x >= 0, y >= 0, z >= 0, y^2 <= x z}, {x, y, z}] yields x=y=z=1/3 in Mathematica. $\endgroup$
    – José Figueroa-O'Farrill
    Mar 29, 2016 at 1:15
  • $\begingroup$ Thanks Josè, but I am not interested in a software solution, rather in methods how to solve it manually.. [Anyway, we expected that solution geometrically :P ] $\endgroup$
    – Nduccio
    Mar 29, 2016 at 1:21
  • $\begingroup$ MathOverflow is for questions of math research interest. It's not clear that your question has a research angle. $\endgroup$
    – Gerry Myerson
    Mar 29, 2016 at 2:00
  • $\begingroup$ I know the policy of MO, probably it is my fault to not have explained the source, and it really comes from a research question. on the other hand, I just wanted to know if the tecniquenes to solve it were standard, at first sight I d say a method with Lagrangian multipliers might be enough, but I didnt know a better approach.. $\endgroup$
    – Nduccio
    Mar 29, 2016 at 9:08
  • $\begingroup$ My point was simply that the method exists, because Mathematica did not solve it numerically. I think this is a standard optimization problem (introduce slack variables for the inequalities,...) $\endgroup$
    – José Figueroa-O'Farrill
    Mar 29, 2016 at 14:23

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