I'm trying to optimize a sum of the form: $$ (\sum_{i} a_i k_i + \sum_{i,j}b_{i}\cdot b_j^{T} c_{i,j} +K)^2, $$ where the $b_i$ is the $i^{th}$ row of a non-zero $3\times 3$ matrix $b$, the $a_i$ are the components of a vectors in $\mathbb{R}^3$, the $c_{i,j}$, $k_i$ and $K$ are non-zero scalars in $\mathbb{R}$ and $\|-\|_2^2$ is the square of the Euclidean norm on $\mathbb{R}^3$.
My question is how do I convert the preceding equation into matrix form and find the vector $a$ and matrix $b$ optimizing it?
I'm quite new to Matrix calculus, so I'm sorry if this question is silly.
Well....if any one of the $k_i$ ... say $k_1$, is nonzero, then you can let all the $b_i$ be zero, pick $a_1 = -K/k_1$, and you get a norm of 0, which is pretty much minimal.
If you're hoping for maximal, then if any one of the $c_{ii}$, say $c_{1,1}$, is positive, you pick $b_1 = \alpha e_1$ and $b_2 = b_3 = 0$, and pick $a_i = 0$ for all $i$. THen by increasing $\alpha$, you can make the target value arbitrarily large.
From these two observations, I have to suspect that you've phrased the question wrong, but perhaps not.