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Matroid Optimization Problems with Monotone Monomials in the Objective

Anja Fischer(anja.fischer***at***mathematik.uni-goettingen.de)
Frank Fischer(frank.fischer***at***uni-kassel.de)
S. Thomas McCormick(tom.mccormick***at***sauder.ubc.ca)

Abstract: In this paper we investigate non-linear matroid optimization problems with polynomial objective functions where the monomials satisfy certain monotonicity properties. Indeed, we study problems where the set of non-linear monomials consists of all non-linear monomials that can be built from a given subset of the variables. Linearizing all non-linear monomials we study the respective polytope. We present a complete description of this polytope. Apart from linearization constraints one needs appropriately strengthened rank inequalities. The separation problem for these inequalities reduces to a submodular function minimization problem. These polyhedral results give rise to a new hierarchy for the solution of matroid optimization problems with polynomial objectives. Finally, we consider extensions of our results and give suggestions for future work.

Keywords: Polynomial 0-1 Programming, Polyhedral Combinatorics, Matroids, Complete Description, Hierarchy

Category 1: Combinatorial Optimization (Graphs and Matroids )

Category 2: Combinatorial Optimization (Polyhedra )

Category 3: Integer Programming (0-1 Programming )

Citation: NAM Preprint 2017-6, University of Goettingen, August 2017

Download: [PDF]

Entry Submitted: 08/24/2017
Entry Accepted: 08/24/2017
Entry Last Modified: 08/24/2017

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