Recent combinatorial algorithms for linear programming can also be applied to certain nonlinear problems. We call these Generalized Linear Programming, or GLP, problems. We connect this class to a collection of results from combinatorial geometry called Helly-type theorems. We show that there is a Helly-type theorem about the constraint set of every GLP problem. Given a family H of sets with a Helly-type theorem, we give a paradigm for finding whether the intersection of H is empty, by formulating the question as a GLP problem. This leads to many applications, including linear expected time algorithms for finding line transversals and minimax hyperplane fitting. Our applications include GLP problems with the surprising property that the constraints are non-convex or even disconnected. 1
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