# selecting a good upper bound on quadratic form in presence of unknown PD matrix

I have a cost function that is $$J=\text{Trace} [\ (\ I-LC)\ KQK^T(\ I-LC)\ ^T ]\$$

where $Q$ is a unknown positive definite matrix, $K$ and $C$ are full rank $n\times n$ and $m\times n$ matrices, respectively. Objective is to find $L$ such that minimizes $J$. Because $Q$ is unknown, I can't use it in minimization, so I tried to minimize upper bound of $J$. We can consider two upper bounds $$J\leq \lambda_{max}(Q) \text{Trace} [\ (\ I-L_1C)\ KK^T(\ I-L_1C)\ ^T ]\$$

$$J\leq \lambda_{max}(KQK^T) \text{Trace} [\ (\ I-L_2C)\ (\ I-L_2C)\ ^T ]\$$ $\lambda_{max}(Q)$ and $\lambda_{max}(KQK^T)$ are unknown, but non-negative, so the only way is to minimizing Trace parts. $L_1$ and $L_2$ that minimize those are $$L_1=KK^TC^T(\ CKK^TC^T)\ ^{-1}$$ $$L_2=C^T(\ CC^T)\ ^{-1}$$

Now it makes sense that $$\lambda_{max}(Q) \text{Trace} [\ (\ I-L_1C)\ KK^T(\ I-L_1C)\ ^T ]\ \leq \lambda_{max}(KQK^T) \text{Trace} [\ (\ I-L_2C)\ (\ I-L_2C)\ ^T ]\$$ but I can't prove this.

what do you think?

Thanks

P.S: A constraint for $Q$ that can be considered is $$\begin{split} Q_{ii}&>|Q_{ij}|\qquad&\text{for}\qquad i\not= j\\ Q_{ii}&=Q_{jj}\qquad&\text{for}\qquad i\not= j \end{split}$$ and $n\gg m$, $C=[\ I_m\quad 0]\$

• What do you mean, $Q$ is unknown? It is another degree of freedom in the optimization? Or you need to minimize something like the supremum of $J$ over all possible $Q$? – user7530 Jan 2 '14 at 21:11
• second one, I need to minimize supremum of $J$ over all symmetric positive definite $Q$. – user118748 Jan 3 '14 at 4:42
• Is there no constraint on $Q$ except being SPD? Otherwise, I can't see why such an optimization problem would have a solution except in the very special case that you can zero-out $I-LC$. – Algebraic Pavel Jan 3 '14 at 12:21
• Edited question. – user118748 Jan 3 '14 at 12:43