Optimization KroneckerNow I have to solve a optimization problemu=∑p=1Ph1,p⊗h2,p.&nbsp;minh1,p,h2,puTRu,with a iterative algorithm, where R is a correlation matrix.

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Optimization KroneckerNow I have to solve a optimization problemu=∑p=1Ph1,p⊗h2,p.&amp;nbsp;minh1,p,h2,puTRu,with a iterative algorithm, where R is a correlation matrix.

ubafumene42h · Accepted Answer

First, minimization of the Rayleigh quotient minu(uTRuuTu) is a well-known problem whose solution is the eigenvector associated with λmin of the R matrix.Second, any matrix can be expanded as a sum of Kronecker productsX=∑k=1rAk⊗Bkif the dimensions of (X,Ak,Bk) are compatible (which is guaranteed by the posted question).The number of terms in the expansion is determined by the value of r which the rank of the matrix X after it has been reshaped and its elements permuted. For further details, look for papers by vanLoan &amp;amp; Pitsianis. Better yet, search for Pitsianis&#039; 1997 thesis, which contains Matlab code for the decomposition.The vector u which minimizes the Rayleigh quotient, can therefore be expanded asu=∑k=1rak⊗bkIdentifying (ak,bk)→(h1,k,h2,k), recovers the form of the current question.Therefore, if P≥r, use the full Pitsianis decomposition, otherwise truncating the sum at the Pth term will yield the &quot;nearest Kronecker approximation&quot; (which is another thing to google).

Optimization Kronecker Now I have to solve a optimization problem \begin{equation}u=\sum_{p=1}^{P}h_{1,p}\otimes h_{2,p}.\\\min_{h_{1,p},h_{2,p}}u^TRu,\end{equation}

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