I want to conduct an SVM model-regression (i.e., support vector regression), using a linear kernel function. Does it make sense to perform a cross-validation hyperparameter optimization when the kernel function is linear? If so, what should be the range of values for each hyperparameter in the search?
Thank you
Yes, it makes full sense to use cross-validation to find optimal hyper-parameters values in the case of SVM with a linear kernel.
If anything, the choosing the regularisation parameter $C$ in the Lagrange formulation is analogous to choosing the ridge regularisation parameter in ridge regression. Therefore it is necessary for our training data to be scaled appropriately (usually to mean $0$ and st.dev. $1$). Regarding the actual choice of the $C$, it is generally common to use exponentially growing sequences of $C$; e.g. $C = 2^{-6},2^{-5},\ldots,2^{5},2^{6}$, etc. CV.SE has a nice thread on this matter if one wants to explore this further: "Which search range for determining SVM optimal C and gamma parameters?".
As usεr11852 says, cross validation does make sense for optimizing linear SVMs.
According to Hastie et al.: The Entire Regularization Path for the Support Vector Machine, Journal of Machine Learning Research 5 (2004) 1391–1415, inside the cross validation you don't need to compute one SVM per C value as it is possible to compute the complete path of SVMs with varying Cs together.
If you work in R, package svmpath provides this.
