I have no idea how to solve the following question and will be grateful for any help. Suppose we are given n real numbers y 1 , . . . , y n . Is there any simple way to minimize function f ( y ) = ∑ k = 1 n | y k − y | ?

Question

I have no idea how to solve the following question and will be grateful for any help. Suppose we are given      n    real numbers           y    1    ,  .  .  .  ,      y    n     . Is there any simple way to minimize function      f  (  y  )  =      ∑          k      =      1              n            |        y    k    −  y      |  ?

Scarlet Reid · Accepted Answer

Without loss of generality, suppose that       y    1    ≤      y    2    ≤  ⋯  ≤      y    n  .      ∑          k      =      1        n        |        y    k    −  y      |    =      1    2    [      ∑          k      =      1        n        |        y          n      +      1      −      k        −  y      |    +      ∑          k      =      1        n        |    y  −      y    k        |    ]  ≥      1    2        ∑          k      =      1        n        |        y          n      +      1      −      k        −      y    k        |  Equality holds only when   y  ∈  [      y    k    ,      y          n      +      1      −      k        ] or   [      y          n      +      1      −      k        ,      y    k    ] for arbitrary   k.If   n is even, the minimize attains when   y  ∈  [      y                  n        2              ,      y                            n          +          1                2              ].If   n is odd, the minimize attains when   y  =      y                            n          +          1                2            .

I have no idea how to solve the following question and will be grateful for any help. Suppose we are

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