  Sharir and Welzl introduced an abstract framework for optimization problems,
called LP-type problems or also generalized linear programming problems, which
proved useful in algorithm design. We define a new, and as we believe, simpler
and more natural framework: violator spaces, which constitute a proper
generalization of LP-type problems. We show that Clarkson's randomized
algorithms for low-dimensional linear programming work in the context of
violator spaces. For example, in this way we obtain the fastest known algorithm
for the P-matrix generalized linear complementarity problem with a constant
number of blocks. We also give two new characterizations of LP-type problems:
they are equivalent to acyclic violator spaces, as well as to concrete LP-type
problems (informally, the constraints in a concrete LP-type problem are subsets
of a linearly ordered ground set, and the value of a set of constraints is the
minimum of its intersection).
