The problem that you are describing can be solved by latent class regression, or cluster-wise regression, or it's extension mixture of generalized linear models that are all members of a wider family of finite mixture models, or latent class models.
It's not a combination of classification (supervised learning) and regression per se, but rather of clustering (unsupervised learning) and regression. The basic approach can be extended so that you predict the class membership using concomitant variables, what makes it even closer to what you are looking for. In fact, using latent class models for classification was described by Vermunt and Magidson (2003) who recommend it for such pourpose.
AdditionalReferences and additional resources
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Wedel, M. and Kamakura, W.A. (2001). Market Segmentation – Conceptual
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Leisch, F. (2004). Flexmix: A general framework for finite mixture
models and latent glass regression in R. Journal of Statistical
Software, 11(8), 1-18.
Grun, B. and Leisch, F. (2008). FlexMix version 2: finite mixtures
with concomitant variables and varying and constant parameters.
Journal of Statistical Software, 28(1), 1-35.
McLachlan, G. and Peel, D. (2000). Finite Mixture Models. John Wiley & Sons.
Dayton, C.M. and Macready, G.B. (1988). Concomitant-Variable
Latent-Class Models. Journal of the American Statistical Association,
83(401), 173-178.
Linzer, D.A. and Lewis, J.B. (2011). poLCA: An R package for
polytomous variable latent class analysis. Journal of Statistical
Software, 42(10), 1-29.
McCutcheon, A.L. (1987). Latent Class Analysis. Sage.
Hagenaars J.A. and McCutcheon, A.L. (2009). Applied Latent Class
Analysis. Cambridge University Press.
Vermunt, J.K., and Magidson, J. (2003). Latent class models for
classification. Computational Statistics & Data Analysis, 41(3),
531-537.
