I seem not to comprehend fully what you meant by classification of missing data. There are different algorithms for classification problems as you rightly mentioned 2. Others include SVM, RF, PLS-DA, SIMCA etc. Dealing with missing data is a separate problem that you may want to tackle before trying your classifier(s). You may try your classifiers after dealing with your missing values to various degrees. Some ways of handling missing values include replacement by small values, total exclusion of the missing values, replacement by mean, median or minimum values. Also you can estimate for missing values using procedures like KNN, PPCA, BPCA, and SVD. You just need to start from any or some of those methods and test your classifier on the preprocessed data. In the long run, you may discover solving your problem may not be in using new classifier but in dealing with your missing values appropriately. Wishing you all the best.
Data imputation is always tricky business. Mean imputation can be problematic because you are shedding away some of the inherent variance in the data set by injecting more observations about the mean. Columbia has a good write up on imputation. Many times a domain expert can be great assistance when deciding how to, or if you should fill in missing data. Asking the question, why is there data missing is very important. Was this a sensor error, or was there no data for a dat range, etc. These question can help guide you towards not imparting biases and noise into your data set. Hope this helps! Cheers, kyle