If we have that the contours of a response surface are elliptical and the response is given by the following function: exp ⁡ ( − ( w 2 + 1 4 l 2 − 1 4 ⋅ w ⋅ l ) ) then if we maximize this function w.r.t l holding w fixed at 1/2.And if we call the maximizer l-star, then holding l-star fixed, maximize over w. How to show that the overall max isn't achieved?My approach: I got the partial of the above function w.r.t. l and then tried to evaluate it at 1/2, but got stuck. It most likely will involve some analysis of Hessians.

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If we have that the contours of a response surface are elliptical and the response is given by the following function:      exp    ⁡          (      −              (                  w          2                +                  1          4                          l          2                −                  1          4                ⋅        w        ⋅        l        )            )      then if we maximize this function w.r.t l holding w fixed at 1/2.And if we call the maximizer l-star, then holding l-star fixed, maximize over w. How to show that the overall max isn&#039;t achieved?My approach: I got the partial of the above function w.r.t. l and then tried to evaluate it at 1/2, but got stuck. It most likely will involve some analysis of Hessians.

trajeronls · Accepted Answer

The exponential function does not actually matter, because it is strictly increasing. The maxima and minima of   exp  ⁡  (  f  (  w  ,  l  )  ) are attained (or not attained) precisely at the same points as maxima and minima of   f  (  w  ,  l  ) itself. One advantage of working with   f  (  w  ,  l  )  =  −  (      w    2    +      l    2        /    4  −  l      /    2  ) is that its derivatives are simpler than for       e    f  .And another advantage that we don&#039;t even need calculus: it&#039;s a quadratic polynomial in which we can complete the square.                    (1)                    −        (                  w          2                +                  l          2                          /                4        −        l                  /                2        )        =        −        (                  w          2                +        (        l        −        1                  )          2                          /                4        −        1                  /                4        )            To maximize −(…), we minimize the content of the parentheses. The smallest       w    2   can be is 0, at   w  =  0. The smallest   (  l  −  1      )    2        /    4 can be is 0, at   l  =  1. Therefore, at   w  =  0,   l  =  1 the global maximum of (1) is attained, and it is equal to 1/4.Consequently,       e    f   has maximum value       e          1              /            4      .

If we have that the contours of a response surface are elliptical and the response is given by the f

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