I need help with the following optimization problem max α ln ⁡ ( x ( 1 − y 2 ) ) + ( 1 − α ) ln ⁡ ( z )where the maximization is with respect to x , y , z, subject to α x + ( 1 − α ) z = C 1 α y x ( x + γ ) − α x = C 2 where 0 ≤ α ≤ 1, γ &gt; 0, and x , z ≥ 0, and | y | ≤ 1.Generally, one can substitute the constraints in the objective function and maximize with respect to one parameter. The problem is that in this way things become algebraically complicated, and I believe that there is a simple solution.

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I need help with the following optimization problem  max    α  ln  ⁡  (  x  (  1  −      y    2    )  )  +  (  1  −  α  )  ln  ⁡  (  z  )where the maximization is with respect to   x  ,  y  ,  z, subject to                    α        x        +        (        1        −        α        )        z                            =                  C          1                                    α        y                  x          (          x          +          γ          )                −        α        x                            =                  C          2                    where   0  ≤  α  ≤  1,   γ  &amp;gt;  0, and   x  ,  z  ≥  0, and       |    y      |    ≤  1.Generally, one can substitute the constraints in the objective function and maximize with respect to one parameter. The problem is that in this way things become algebraically complicated, and I believe that there is a simple solution.

agentbangsterfhes2 · Accepted Answer

You should use Lagrange multipliers, write out the Lagrangian, differentiate it w.r.t.   x,  y and   z and set them to zero.Your Lagrangian for this problem would be:      L    (  x  ,  y  ,  z  )  =  α  l  n  (  x  (  1  −      y    2    )  )  +  (  1  −  α  )  l  n  (  z  )  +      λ    1    (  α  x  +  (  1  −  α  )  z  −      C    1    )  +      λ    2    (  α  y      x    (    x    +    γ    )    −  α  x  −      C    2    )You need to set             ∂              L                    ∂      x        =            ∂              L                    ∂      y        =            ∂              L                    ∂      z        =  0 and eliminate       λ    1   and       λ    2   to get your optimal       x    ∗  ,       y    ∗   and       z    ∗  .

agrejas0hxpx · Accepted Answer

Use the second constraint to eliminate   y and the first constraint to eliminate   z from the expression that you want to maximize. The result is a function of   x, in fact a linear combination of three terms of the form   log  ⁡  (      A    k    x  +      B    k    ).The complete picture also depends on the values of       C    1   and       C    2  , about which we have been told nothing.

I need help with the following optimization problem <mo movablelimits="true" form="prefix">max

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