[?] Perform optimization with a strange constraint?

Questions
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1

Consider the following code:

Table[NMaximize[{NIntegrate[(a*(1 - 0.25*z - (1 - 0.25)*x)*x + 
           a*(0.25*z - y + (1 - 0.25)*x)*y)/b, {a, 0, b}], 
       z = Min[1, (x - y*0.3)/(1 - 0.3)], b*(1 - y) <= 1, 
       0 <= y <= x <= 1}, {x, y}], {b, 1, 1.5, 0.1}]

I am trying to numerically solve an optimization problem (for x and y, for some b values). The problem has an integration in it. Interestingly, there is another variable z, which is a function of x and y, but cannot be greater than 1 (therefore I use Min function).
Integration have issues because of the Min function. Any ideas?

Cheers,
Ovunc

2

Assuming I didn’t make any stupid mistakes:

  • NIntegrate → Integrate, evaluate by itself first, get analytic expression
  • replace z with Min expression in there
  • replace last condition with y<=x, x<=1 (0<=y is covered by the one containing b for the given values)

Then I end up with:

In[11]:= Table[
 NMaximize[{(b x)/2 - 0.375` b x^2 + 0.375` b x y - (b y^2)/2 - 
    0.125` b x Min[1, 1.4285714285714286` (x - 0.3` y)] + 
    0.125` b y Min[1, 1.4285714285714286` (x - 0.3` y)], 
   b*(1 - y) <= 1, y <= x, x <= 1}, {x, y}], {b, 1., 1.5, 0.1}]

Out[11]= {{0.161458, {x -> 0.645833, 
   y -> 0.354167}}, {0.177604, {x -> 0.645833, 
   y -> 0.354167}}, {0.19375, {x -> 0.645833, 
   y -> 0.354167}}, {0.209896, {x -> 0.645833, 
   y -> 0.354167}}, {0.226042, {x -> 0.645833, 
   y -> 0.354167}}, {0.242187, {x -> 0.645833, y -> 0.354167}}}

Hope this helps,
Peter

3

IGNORE THE FOLLOWING - IT ACTUALLY WORKED AFTER A SMALL CHEAT:)

This worked very well, thanks so much Peter!
Only issue, when the problem is bigger and also the integration bounds have “z” in them - it does not run at all.

oint3 = Integrate[(a*x*(1 - 0.25*z - (1 - 0.25)*x) + 
a*y*(0.25*z - y + (1 - 0.25)*x))/b, {a, 0, 1/(1 - y)}] + 
Integrate[(a*x*(1 - 0.25*z - (1 - 0.25)*x) + 
y*(1 - a*(1 - 0.25*z - (1 - 0.25)*x)))/b, {a, 1/(1 - y), 
1/(1 - (1 - 0.25)*x - 0.25*z)}] + 
Integrate[x/b, {a, 1/(1 - (1 - 0.25)*x - 0.25*z), b}]

Let’s say we maximize the sum of three integrals with the constraints

0 <= y <= x <= 1, 
   b*(1 - (1 - 0.25)*x - 0.25*Min[1, (x - y*0.1)/(1 - 0.1)]) >= 
    1}, {x, y}], {b, 1, 6, 0.1}]

The same idea just cannot be used. Any ideas?
I extremely appreciate your help in this.