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On max-k-sums

Michael Todd (mjt7***at***cornell.edu)

Abstract: The max-$k$-sum of a set of real scalars is the maximum sum of a subset of size $k$, or alternatively the sum of the $k$ largest elements. We study two extensions: First, we show how to obtain smooth approximations to functions that are pointwise max-$k$-sums of smooth functions. Second, we discuss how the max-$k$-sum can be defined on vectors in a finite-dimensional real vector space ordered by a closed convex cone.

Keywords: maxima, convexity, smoothing, symmetric cones

Category 1: Applications -- Science and Engineering (Basic Sciences Applications )

Category 2: Convex and Nonsmooth Optimization (Convex Optimization )

Category 3: Linear, Cone and Semidefinite Programming (Other )

Citation: manuscript, School of Operations Research and Information Engineering, Cornell University, Ithaca, NY, September 2016 To appear in Mathematical Programming. The final publication is available at link.springer.com at https://link.springer.com/article/10.1007/s10107-017-1201-0

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Entry Submitted: 09/03/2016
Entry Accepted: 09/03/2016
Entry Last Modified: 11/21/2017

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