I would like to know if it is possible to solve the following nonlinear constrained optimization problem and find how the optimal solution varies with $C$ and $\beta$:
$\max_{x,y}\beta F(x)-d(2-F(x)-F(y)),s.t$
$\int_{-\infty}^{x}uf(u)du+\int_{-\infty}^{y}vf(v)dv\leqslant C$,
where $F(.)$ and $f(.)$ are CDF and PDF of standard normal distributions, respectively, and $d(.)$ is assumed to be sufficiently convex so that interior solution exists.
Thanks!