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Since the $f_i(\theta,\phi)$ are linear combinations of spherical harmonics, we can write $$ \mathbf{f} = F \mathbf{Y} $$ where $\mathbf{Y}$ is a vector of the orthonormalized spherical harmonics - i.e.: $$ \int d\Omega Y_l^m Y_{l'}^{m'*} = \delta_{ll'} \delta_{mm'} $$ So the integral becomes, $$ \int d\Omega \mathbf{f}^{\dagger} (B^{\dagger} + B) \mathbf{f} ...


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I'm going to assume that you optimize over the locations $x_i$. Then this is most easily reformulated via slack variables as follows: $$ \min_{x_i,s_i} s_i^2 \\ \text{so that}\quad s_i \le g(x_i) \\ \qquad\quad s_i \le 0 $$ This is not a convex problem because the feasible region described by these constraints is not convex. You can see this by just ...


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Assuming $B$ is not angle-dependent, you know the coefficients of each $f_i$ a priori, and $\mathbf{f}$ can only be evaluated by summing up the spherical harmonics expansion, simplify your problem by making use of the orthogonality of spherical harmonics instead of numerically integrating. Let $f_i(\theta,\phi)=\sum_{lm}f_{i,lm}Y_l^m(\theta,\phi)$. Then your ...


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