Abstract
Linear optimization problems (LPs) with a very large or even infinite number of constraints frequently appear in many forms in machine learning. A linear program with m constraints can be written as where I assume for simplicity that the domain of x is the n dimensional probability simplex . Optimization problems with an infinite number of constraints of the form , for all j∈J, are called semi-infinite, when the index set J has infinitely many elements, e.g. J=ℝ. In the finite case the constraints can be described by a matrix with m rows and n columns that can be used to directly solve the LP. In semi-infinite linear programs (SILPs) the constraints are often given in a functional form depending on j or implicitly defined, for instance by the outcome of another algorithm.
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