multivariable bilevel optimization with integration

[Jong-Chyi Frederick Su]

Hi all,

I am solving a problem which has bilevel optimization with integration involved. The problem is:

(w^*,k^*) = argmax_{w\geq c,k\geq 0} (w-c)E_{\theta_1- \theta_2}[E_{\epsilon=-\delta - \delta}[min(\theta+\epsilon-p^*,k)]]

where 

p^*(\theta,w,k) = argmax_{p\geq w}E_{\epsilon}[min(\theta+\epsilon-p,k)]

Here is my code:

OO = (c-w) * (int(  0.02*max(min(theta+x-p,k),0) ,[x theta], [double(-delta) double(theta_1)], [double(delta) double(theta_2)] ));

CO = [w<= 1000, k<=1000, w >= c, k >= 0, p<=1000, p >= 0 , theta >= theta_1, theta <= theta_2, x >= -delta, x <= delta];

    

OI = (w-p) * (int( 0.02*max(min(theta+x-p,k),0), x, -delta, delta));

CI = [w<= 1000, k<=1000, w >= c, k >= 0, p<=1000, p >= 0 , theta >= theta_1, theta <= theta_2, x >= -delta, x <= delta];

solvebilevel(CO,OO,CI,OI,[p w k])

However, I can't get the result. Can anyone help me about this? Thanks!!!

[Jong-Chyi Frederick Su]

The formula is typed in latex format.

[Johan Löfberg]

YALMIPs bilvel framework assumes the inner problem is an LP or convex QP. Including an "int" operator most certainly violates that (apart from the fact that the int operator only should be applied on polynomial objects, it will not work with higher-level operators such as min)

[Jong-Chyi Frederick Su]

Thanks!!

I'll try another way to solve this!