In the Mathematics Subject Classification scheme (MSC2010), the field of "Set-valued and variational analysis" is coded by "49J53".
[2] While this area of mathematics has a long history, the first use of the term "Variational analysis" in this sense was in an eponymous book by R. Tyrrell Rockafellar and Roger J-B Wets.
[1][failed verification] A classical result is that a lower semicontinuous function on a compact set attains its minimum.
Results from variational analysis such as Ekeland's variational principle allow us to extend this result of lower semicontinuous functions on non-compact sets provided that the function has a lower bound and at the cost of adding a small perturbation to the function.
The ideas of these classical results can be extended to nondifferentiable convex functions by generalizing the notion of derivative to that of subderivative.