I am currently reading this paper : https://www.di.ens.fr/~fbach/skm_icml.pdf about Multi-instance kernels, and in section 2.1 (page 2) it is written the coefficients $\eta_j$ are recovered as Lagrange multipliers for the constraint ... where $\eta_j$ are the "weights" for the subkernels. I don't really get what this means. The way I understand it is that I should consider the Lagrangian:
$$\mathcal L(\zeta, \pmb \alpha, \pmb \eta) = \zeta - 2 \cdot \mathbf 1^\top \pmb \alpha + \sum_j\eta_j\left(\pmb \alpha^\top D(y)K_jD(y)\pmb\alpha - \frac {\text{Tr}K_j}{c}\zeta\right)$$
under the remaining constraints and then solve for $(\zeta, \pmb \alpha, \pmb \eta)$. But this turns the initial linear problem with quadratic constraint into a cubic problem with linear constraints and this does not seem like something I want to solve...
Am I just misunderstaing the word recover? Is it that I need to solve both the primal and the dual and then look at the variables in the dual?