Let $k\in\mathbb N$ and $b_j,c_{ij}>0$ for $i,j\in\left\{1,\ldots,k\right\}$. I want to solve the following problem in $v_i,w_j$:
$$\begin{array}{ll} \text{minimize} & \sum_{i=1}^k\sum_{j=1}^kb_jc_{ij}v_iw_j^2\\ \text{subject to} & \sum_{i=1}^kv_i=\sum_{j=1}^kw_j=1\\ & v_i, w_j\ge0\end{array}\tag1$$
I am aware of the method of Lagrange multipliers, but I wasn't able to figure out how we might apply it here. So, what do we need to do?
EDIT: Ignoring the inequality constraints for a moment we could apply the Lagrange multiplier theorem as follows: Let $f(v,w):=\sum_{i=1}^k\sum_{j=1}^kb_jc_{ij}v_iw_j^2$ and $h(v,w):=\left(\sum_{i\in I}v_i-1,\sum_{i\in I}w_i-1\right)^T$ for $v,w\in\mathbb R^I$ and assume $(v,w)$ is a minimizer of $(1)$ without the nonnegativity constraint. Then there are $(\eta,\lambda)\in\mathbb R\times\mathbb R^2$ with
- $(\eta,\lambda)\ne0$
- $\eta\in\{0,1\}$
- $\nabla(f+\langle\lambda,h\rangle)(v,w)=0$
Now note that $$\nabla\langle\lambda,h\rangle(v,w)=(\lambda_1,\ldots,\lambda_1,\lambda_2,\ldots,\lambda_2)^T\tag2$$ and $$\nabla f(v,w)=\left(\sum_{j\in I}b_jc_{1j}w_j^2,\ldots,\sum_{j\in I}b_jc_{|I|j}w_j^2,2w_1b_1\sum_{i\in I}c_{i1}v_i,\ldots,2w_{|I|}b_{|I|}\sum_{i\in I}c_{i{|I|}}v_i\right)^T.\tag3$$ If $\eta=0$, we would immediately obtain $\lambda=0$, which is impossible by 1. If $\eta=1$ we obtain a system of equations. How should we solve it?