  A classic result of Lenstra [Math.~Oper.~Res.~1983] says that an integer
linear program can be solved in fixed-parameter tractable (FPT) time for the
parameter being the number of variables. We extend this result by incorporating
non-decreasing piecewise linear convex or concave functions to our (mixed)
integer programs. This general technique allows us to establish parameterized
complexity of a number of classic computational problems. In particular, we
prove that Weighted Set Multicover is in FPT when parameterized by the number
of elements to cover, and that there exists an FPT-time approximation scheme
for Multiset Multicover for the same parameter. Further, we use our general
technique to prove that a number of problems from computational social choice
(e.g., problems related to bribery and control in elections) are in FPT when
parameterized by the number of candidates. For bribery, this resolves a nearly
10-year old family of open problems, and for weighted electoral control of
Approval voting, this improves some previously known XP-memberships to
FPT-memberships.
