# Maping non-linear constraint to linear subspace

What I will ask, more than a solution, is the correct definition of my problem and directions to find the solution.

I have a set of linear equations, e.g.:

\begin{eqnarray*} d_1 =& L_1 - 9\,m_1 - 9\,m_2 \\ d_2 =& x_1 + 3\,m_1 + 3\,m_2 \\ d_3 =& y_1 \\ d_4 =& L_2 - 4\,m_2 \\ d_5 =& x_2 + 2\,m_2 \\ d_6 =& y_2 \end{eqnarray*}

where $d_1,d_2,...,d_6$ are linear combinations of $L_1,x_1,y_1,m_1,L_2,x_2,y_2,m_2$.

I can estimate $d_1,d_2,...,d_6$ values, but not the other parameters. $L_1,x_1,y_1,m_1,L_2,x_2,y_2,m_2$ are physical parameters which have physical nonlinear constraints:

\begin{eqnarray*} m_1 > 0 \\ L_1 m_1 - x_1^2 - y_1^2 > 0 \\ m_2 > 0 \\ L_2 m_2 - x_2^2 - y_2^2 > 0 \end{eqnarray*}

I would like to rewrite constraints in terms of $d_1,d_2,...,d_6$ only.

I.e., I would like to find nonlinear constrains over $d_1,d_2,...,d_6$ parameters (only), so that when they are verified it means that there is at least one $L_1,x_1,y_1,m_1,L_2,x_2,y_2,m_2$ solution (it doesn't matter what) which verify the former constraints. If the new constraint are not verified it must mean that no $L_1,x_1,y_1,m_1,L_2,x_2,y_2,m_2$ solution exists.

Here I presented a particularly small example, for it I was already able to find the constraints doing manual equation manipulation (can put them here). However, I have problems with up to 70 linear equations and 30 constraints.

What I need is a systematic method to write the constrains over $d_1,d_2,...,d_6$.

Now the questions:

• What kind of problem do I have?
• Which mathematical fields shall I study, and which directions must I follow?
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