max min [ α x 1 , β x 2 , γ x 3 ] s.t. λ 1 x 1 + λ 2 x 2 + λ 3 x 3 = c , α , β , γ , λ i , c are constantsWell, that function is not differentiable , so what methods can be applied to solve for for the optimal values of x 1 , x 2 and x 3 ? Is knowledge of the λ ′ s and c necessary, to at least some degree, or does a general approach/solution exist?

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max     min  [  α      x    1    ,  β      x    2    ,  γ      x    3    ]        s.t.         λ    1        x    1    +      λ    2        x    2    +      λ    3        x    3    =  c  ,       α  ,  β  ,  γ  ,      λ    i    ,  c     are constantsWell, that function is not differentiable , so what methods can be applied to solve for for the optimal values of       x    1  ,      x    2   and       x    3  ? Is knowledge of the       λ    ′    s and   c necessary, to at least some degree, or does a general approach/solution exist?

upornompe · Accepted Answer

You can reformulate it to be a Linear Program:                    max                            z                            z                            ≤        α                  x          1                                    z                            ≤        β                  x          2                                    z                            ≤        γ                  x          3                                              λ          1                          x          1                +                  λ          2                          x          2                +                  λ          3                          x          3                                    =        c            which you can now feed into a Linear Programming solver and get a answer very easily. A closed-form expression for the optimum probably exists and this can be explored if you write the dual of this problem and use the sign constraints on the variables and parameters.

Sonia Gay · Accepted Answer

When the objective function is not differentiable, a common procedure is the use of subgradient method. Subgradient Methods - Stanford.

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