= Francis Bach
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<Francis Bach> developed a convex-analytic treatment of <submodular set functions> and their applications to <machine learning>. His monograph connects discrete minimization, <polyhedron>[polyhedra], <regularization> and the <Lovász extension>, making relationships between combinatorial and <convex optimization> explicit.<Learning with Submodular Functions: A Convex Optimization Perspective by Francis Bach>[abstract and Chapters2–3]{ref}
For a finite ground <set>, <submodularity> is equivalent to diminishing marginal returns: adding an element to a smaller <set> gives at least as large a gain as adding it to a containing <set>. A <coverage function> illustrates this because objects already covered cannot contribute new coverage. The same structural property can support both direct selection problems and continuous optimization formulations; the modelling objective determines which connection is useful.
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