Maximize a function containing Max

Question

I am trying to maximize a function (involving several Max and Min) in two variables and one parameter with some constraints. After struggling with it, I am trying with smaller examples to understand what I am doing wrong. I write here the three small examples I am working on:

1) 2 variables and 1 parameter:

    Maximize[100 g x - 1/6 y + x, 0 < y < 10 && 0 < x < 10 && g > 0, {x, y}]

which gives the solution:

    10 (1 + 100 g), g>0 
    -\[Infinity], True 

{x -> Indeterminate, y -> Indeterminate}}

2) 2 variables and a Max:

    Maximize[100  x - 1/6 y + Max[0, x], 0 < y < 10 && 0 < x < 10 , {x, y}]

which gives the solution:

   1010, {x -> 10, y -> 0}

3) 2 variables, 1 parameter and a Max:

   Maximize[100 g x - 1/6 y + Max[0, x], g > 0 && 0 < y < 10 && 0 < x < 10 , {x,y}]

which does not solve the problem; just gives this solution:

    Maximize[100 g x - 1/6 y + Max[0, x], g > 0 && 0 < y < 10 && 0 < x < 10 ,{x,y}].

Now my question is: why the combination of case 1) and 2) together does not work?

Thank you very much!

Answer 1

Make the intervals half-closed (or closed)

Assuming[g > 0, 
 Maximize[{100 g x - 1/6 y + x, 0 <= y < 10, 0 < x <= 10}, {x, y}] // 
  Simplify]

(* {10 + 1000 g, {x -> 10, y -> 0}} *)

EDIT:

Assuming[{g > 0, x > 0}, 
 Maximize[{100 g x - 1/6 y + Max[x, 0], 0 <= y <= 10, 
    0 < x <= 10}, {x, y}] // Simplify]

(* {10 + 1000 g, {x -> 10, y -> 0}} *)