Suppose we want to implement an interpolation scheme for some 2D function \(z=f(x,y)\). We assume we have some data points \((x_i,y_j,z_{i,j})\) where the \((x_i,y_j)\) values form a grid. The grid points do not have to be equidistant, but the rows and columns must align:
All these three examples are valid for this approach.
A 2D SOS2 based interpolation scheme can be expressed as:
We force that up to four neighboring elements can be non-zero in the matrix \(\lambda_{i,j}\). This is accomplished by calculating row- and column-sums and then saying up to two consecutive elements of these vectors are nonzero. This scheme is rather simple but it actually works.
Alternative
If we can make the function \(f(x,y)\) separable, we may be able to use a different scheme. E.g. assume \(f(x,y)=x\cdot y\) with \(x,y>0\). We can make this separable by applying a log transformation: \(\log(z)=\log(x)+\log(y)\). We can implement each of the functions \(z’=\log(z),x’=\log(x),y’=\log(y)\) using standard 1D piecewise linear formulations. Then we just add the constraint: \(z’=x’+y’\).
Of course another alternative is using a nonlinear solver.
Reference
- H.P.Williams, “Model Building in Mathematical Programming”, 5th edition, Wiley, 2013.