def delta_Omega(X):
    """
        Compute delta_Omega(X[:, j]) for each X[:, j].
        delta_Omega(x) = sup_{y >= 0} y^T x - Omega(y).

        Parameters
        ----------
        X: array, shape = len(a) x len(b)
            Input array.

        Returns
        -------
        v: array, len(b)
            Values: v[j] = delta_Omega(X[:, j])
        G: array, len(a) x len(b)
            Gradients: G[:, j] = nabla delta_Omega(X[:, j])
        """
    raise NotImplementedError
