I am trying to maximize a function of this type:

$$ max_{x,y} \int_{\phi (x,y)}^1 f(z,y) dz - \int_{\phi (x,y)}^1 g(y) dz - \xi \int^{\phi (x,y)}_0 \int_{\phi (x,y)}^1 g(y)\, dz\, dz$$ Subject to: $$ s.t. \;\;\;\; 0< \phi(x^*,y^*)\leq 1 \;\;\;\; \text{and} \;\;\;\; 0 \leq y \leq 1$$

What method can I use to handle the constraints given that there is a strict inequality sign?



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