I have a function of the form $S=x^{T}W_{1}a_{1}+...+x^{T}W_{n}a_{n}$ where $x,a_{i}\in\mathbb{R}^{d}$, $W_{i}\in\mathbb{R}^{d\times d}$.
I'd like to find $y,b_{1},b_{2},..,b_{n}\in\mathbb{R}^{d}$ solving the maximization $$\max_{y,b_{1},b_{2},..,b_{n}}\lambda_{1}y^{T}W_{1}b_{1}+...+\lambda_{n}y^{T}W_{n}b_{n}+\left|y-x\right|^{2}-\sum\left|a_{i}-b_{i}\right|^{2}$$ Where the lambdas are constants, and all vector norms are bounded by a constant.The soft constraints favor solutions where $y$ is far from $x$, and $a_i$ are near $b_i$. Clearly the solution $x=y$, $a_i=b_i\ \ \forall i$ could be a decent solution, but it may not be the best.
I'd appreciate any ideas on how to approach this. Advice regarding reformulating the constraints to make it easier to solve would also be helpful.
Many thanks!