Machine learning methods for exploring relationships for a continuous response variable I would like to explore a model to predict the value of a continuous response variable, from a set (around 100) of explanatory variables. I do not want to apply PCA like feature reduction, because I want to keep my model maximally informative. 
A straightforward method is to conduct a multiple linear regression on a pre-selected subset of candidate explanatory variables. However, a multiple regression requires 1) no missing measurement from any of the explanatory variables; and 2) linear relationship. In my case, I notice some explanatory variables do show strong linear relationship but with many missing values. Therefore, conducting a multiple regression including all candidate explanatory variables is not feasible. 
My questions are:
1) Are there any machine learning techniques that can handle continuous response variable, but with many NA in many explanatory variables?  The relationships can be non-linear.
2) Is regression tree a proper method to use in my case? 
 A: Random forests (RFs) and gradient boosting machines (GBMs, also called boosted regression trees) can do this. Both are methods based on ensembles of regression trees. I should note that there are certainly other methods that can be used for problems such as this, but I'll focus on these since I am somewhat familiar with them.
In the case of RFs, there are multiple approaches used that depend on the implementation: NAs may either be excluded or handled by imputation (based on the mean/median value, or value of similar points). 
GBMs (perhaps not universally, but in the implementations I've seen) handle NAs in a different way: through surrogate splits. This is explained in Tierney et al. (2015) as: 

[Surrogate split] means that when a value for a variable is missing, and that
  variable needs to be used to determine a split, an alternative
  variable that is highly correlated with the missing variable is used
  to determine the direction of the split.

My impression is that the GBMs do a better job of handling missing data, but I don't have a lot of evidence for that claim. 

Tierney, N. J., Harden, F. A., Harden, M. J., & Mengersen, K. L. (2015). Using decision trees to understand structure in missing data. BMJ Open, 5(6), e007450. http://doi.org/10.1136/bmjopen-2014-007450
