I would like to find
$$\min_{w,b} w_iA_{ij}w_{j} + b_iB_{ij}b_j + 2\alpha_iw_i + 2\beta_ib_i $$
Constrained to: $$ w_i > 0 $$
$$ \sum w_i = 1 $$ $$ b_i=w_i^2 $$
Where $A$ and $B$ are positive semidefinite and $\alpha$ and $\beta$ are arbitrary real vectors
What is the "standard form" for this problem? What is the relevant literature? Are there software packages than readily implement a solution?
The problem would be easy enough if it wasn't for the last constraint.