I am an economist with some math background but not strong enough to solve this. I'm trying to solve: max x f ( x , y ( x ) ) = a ⋅ y ( x ) + b ⋅ g ( x ) where y ( x ) = 1 ( h ( x ) &gt; 0 ) . h ( x ) is a linear function of x and a , b ∈ R are constants. x ∈ X and X is a compact and continuous subset of [ 0 , 1 ]. g ( x ) is a concave and differentiable function.Now, my problem is that the optimal choice of x depends on the value of y( x), which depends on x.I would like to know how to solve for, if possible, a closed-form solution. If it is impossible, what kind of algorithm should I use, and where can I find the essential readings to learn them.EDIT: Thanks Robert for his great answer. Now I would just like to ask this follow-up question:Can this method of solving for the maximum be adopted in a dynamic programming version of this problem? Say I want to maximise V ( x t ) = max x t f ( x t , y t ( x t ) ) + δ V ( x t + 1 ) but the constant a now becomes a function of x t − 1 ? δ is a discount factor.

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I am an economist with some math background but not strong enough to solve this. I&#039;m trying to solve:                              max          x                         f        (        x        ,        y        (        x        )        )        =        a        ⋅        y        (        x        )        +        b        ⋅        g        (        x        )            where                    y        (        x        )        =        1        (        h        (        x        )        &amp;gt;        0        )        .              h  (  x  ) is a linear function of   x and   a  ,  b  ∈      R   are constants.   x  ∈  X and   X is a compact and continuous subset of   [  0  ,  1  ].   g  (  x  ) is a concave and differentiable function.Now, my problem is that the optimal choice of   x depends on the value of   y(  x), which depends on   x.I would like to know how to solve for, if possible, a closed-form solution. If it is impossible, what kind of algorithm should I use, and where can I find the essential readings to learn them.EDIT: Thanks Robert for his great answer. Now I would just like to ask this follow-up question:Can this method of solving for the maximum be adopted in a dynamic programming version of this problem? Say I want to maximise  V  (      x    t    )  =      max                  x        t              f  (      x    t    ,      y    t    (      x    t    )  )  +  δ  V  (      x          t      +      1        ) but the constant   a now becomes a function of       x          t      −      1      ?   δ is a discount factor.

Amir Beck · Accepted Answer

Essentially there are two separate problems to consider:1. maximize   b  g  (  x  ) on   {  x  :  h  (  x  )  ≤  0  }.2. maximize   a  +  b  g  (  x  ) on   {  x  :  h  (  x  )  &amp;gt;  0  }.and then you take whichever of these solutions has the best objective value. To complicate matters, however, there is no guarantee that a maximum is attained in (2), as   {  x  :  h  (  x  )  &amp;gt;  0  } is not closed. If   a  &amp;gt;  0 and the maximum of   a  +  b  g  (  x  ) on   {  x  :  h  (  x  )  ≥  0  } occurs only at a point where   h  (  x  )  =  0, the problem does not have an optimal solution.

I am an economist with some math background but not strong enough to solve this. I'm trying to solve

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