Minimizing a Submodular Function on a Lattice
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TL;DR
General conditions under which a collection of optimization problems, with the objective function and the constraint set depending on a parameter, has optimal solutions that are an isotone function of the parameter are given.
Abstract
This paper gives general conditions under which a collection of optimization problems, with the objective function and the constraint set depending on a parameter, has optimal solutions that are an isotone function of the parameter. Relating to this, we present a theory that explores and elaborates on the problem of minimizing a submodular function on a lattice.