# Combining multiple constraints in optimization

Let's say we have two equations for a pose $\textbf{R},\textbf{t}$ optimization problem.

For ICP constraint $$\textbf{e}{_{I}}=\textbf{n}^\top(\textbf{p}_d-(\textbf{R}\textbf{p}_s+\textbf{t}))$$

For epipolar constraint $$\textbf{e}{_{c}}=\textbf{u}_s^\top[\textbf{t}]_{\times}\textbf{R}\textbf{u}_d$$

Then, for the optimization we define an objective function as follows and find $\textbf{R},\textbf{t}$ that minimize it. $$\mathbf{J}=\alpha\frac{1}{2}\sum_{v=1}^{V}{\textbf{e}{_{I_v}}^\top\boldsymbol{\Sigma}_{I_v}^{-1}\textbf{e}{_{I_v}}}+\beta{\frac{1}{2}}\sum_{j=1}^{J}{\textbf{e}{_c{_j}}^\top\boldsymbol{\Sigma}_{c_j}^{-1}\textbf{e}{_{c_j}}}$$

My question is that what is the common practics to decide the weights $\alpha,\beta$. The scales of each constraint are too different. If we do not set it properly, the one with very small scale will be ignored. I have been finding these values manually, but I believe that there are better ways.