let T ≥1 be some finite integer, solve the following maximization problem.Maximize ∑ t = 1 T ( 1 2 ) x t subject to ∑ t = 1 T , x t ≤ 1, x t ≥ 0, t=1,...,TI have never had to maximize summations before and I do not know how to do so. Can someone show me a step by step break down of the solution?

Question

let   T  ≥1 be some finite integer, solve the following maximization problem.Maximize       ∑          t      =      1        T  (      1    2  )            x      t       subject to       ∑          t      =      1              T      ,       x    t    ≤  1,       x    t    ≥  0, t=1,...,TI have never had to maximize summations before and I do not know how to do so. Can someone show me a step by step break down of the solution?

laure6237ma · Accepted Answer

Use the method of Lagrange Multipliers.Denote your objective function by   f  (  x  )  =      ∑          t      =      1        T    (      1    2        )    t              x      t      .Since your constraint is on the sum of the x variables, an optimal solution       x    ∗   will have a gradient which is parallel to the vector   (  1  ,  1  ,  …  ,  1  ), i.e. We seek a point where   ∇  f  (      x    ∗    )  =  λ  (  1  ,  1  ,  …  ,  1  ).Computing the gradient of   f, we get   (      1          4                        x          1                      ,      1          8                        x          2                      ,  …  ,      1                  2                  T          +          1                                      x          T                      )  .Solve for which       x    ∗   causes the gradient to have all its coordinates equal. In other words, find values for       x    1    ,      x    2    ,  …  ,      x    T   so that      1          4                        x          1                      =      1          8                        x          2                      =  …  =      1                  2                  T          +          1                                      x          T                    Then you will have your optimal solution.

let T ≥ 1 be some finite integer, solve the following maximization problem.

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