Calculates the objective function and its gradient for estimating the \(M\)-envelope of span(\(U\)), where \(M\) is positive definite and \(U\) is positive semi-definite.
FGfun(Gamma, M, U)\(\Gamma\) matrix in the envelope objective function. A \(p\)-by-\(u\) matrix.
The \(p\)-by-\(p\) positive definite matrix \(M\) in the envelope objective function.
The \(p\)-by-\(p\) positive semi-definite matrix \(U\) in the envelope objective function.
The value of the objective function at Gamma.
The value of the gradient function at Gamma.
The generic objective function \(F(\Gamma)\) and its gradient \(G(\Gamma)\) are listed below for estimating \(M\)-envelope of span(\(U\)). For the detailed description, see Cook, R. D., & Zhang, X. (2016).
$$F(\Gamma)=\log|\Gamma^T M \Gamma|+\log| \Gamma^T(M+U)^{-1}\Gamma|$$ $$G(\Gamma) = dF/d \Gamma = 2 M \Gamma (\Gamma^T M \Gamma)^{-1} + 2 (M + U)^{-1} \Gamma (\Gamma^T (M + U)^{-1} \Gamma)^{-1}$$
Cook, R.D. and Zhang, X., 2016. Algorithms for envelope estimation. Journal of Computational and Graphical Statistics, 25(1), pp.284-300.