I have a common optimization problem $$\arg\min_A \text{tr}( A^TWA),$$ where $W$ is a positive semi-definite matrix, and $A$ is the matrix to be optimized.
If $A$ is completely unknown, with some scaling constraints, one can reduce this problem into an eigenproblem.
What if $A$ is partially given? That is, I know some entries of $A$ and wish to optimize wrt the unknown entries.
Is there a closed-form solution? What method should I use for this optimization? Pointers to relevant papers/books will be very much appreciated.
After some research, I think this is a quadratic programming problem? But I've only seen quadratic problem that solves for a vector $x$. What if I want to solve for a matrix $A$? Is this still a quadratic programming problem?
Without any constraint, you can solve it explicitly. Just write it as $d + c^Tx + x^TQx$ and the solution is given by $-\frac{1}{2}Q^{-1}c$ (assuming $Q$ psd, similar otherwise).
To get it in standard form, write $A = A_0 +\sum_{i=1}^n A_i x_i$ and insert into objective $\text{trace} A^TWA = \text{trace} (A_0 +\sum_{i=1}^n A_i x_i)^TW(A_0 +\sum_{i=1}^n A_i x_i)$. From this you can extract the vectorized model, e.g., $Q_{ij} = \text{trace}~A_i^T W A_j$ etc.
For example, if $A = \begin{pmatrix} x_1 & 4x_1+x_2\\0 & x_2\end{pmatrix}$ then $A_1 = \begin{pmatrix} 1 & 4\\0 & 0\end{pmatrix}$ and $A_2 = \begin{pmatrix} 0 & 1\\0 & 1\end{pmatrix}$