Calculus of convex polyhedra and polyhedral convex functions by utilizing a multiple objective linear programming solver

The article deals with operations defined on convex polyhedra or polyhedral convex functions. Given two convex polyhedra, operations like Minkowski sum, intersection and closed convex hull of the union are considered. Basic operations for one convex polyhedron are, for example, the polar, the conica...

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Bibliographic Details
Published in:Optimization Vol. 68; no. 10; pp. 2039 - 2054
Main Authors: Ciripoi, Daniel, Löhne, Andreas, Weißing, Benjamin
Format: Journal Article
Language:English
Published: Philadelphia Taylor & Francis 03.10.2019
Taylor & Francis LLC
Subjects:
Affine transformations
Calculus
Computational geometry
Convex analysis
Convexity
Hulls
Linear programming
Mathematical analysis
Multiple objective analysis
multiple objective linear programming
Polyhedra
polyhedral convex analysis
polyhedral set
Polyhedron
polyhedron computations
Representations
ISSN:0233-1934, 1029-4945
Online Access:Get full text
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Summary:The article deals with operations defined on convex polyhedra or polyhedral convex functions. Given two convex polyhedra, operations like Minkowski sum, intersection and closed convex hull of the union are considered. Basic operations for one convex polyhedron are, for example, the polar, the conical hull and the image under affine transformation. The concept of a P-representation of a convex polyhedron is introduced. It is shown that many polyhedral calculus operations can be expressed explicitly in terms of P-representations. We point out that all the relevant computational effort for polyhedral calculus consists in computing projections of convex polyhedra. In order to compute projections we use a recent result saying that multiple objective linear programming (MOLP) is equivalent to the polyhedral projection problem. Based on the MOLP solver bensolve a polyhedral calculus toolbox for Matlab and GNU Octave is developed. Some numerical experiments are discussed.
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ISSN:0233-1934
1029-4945
DOI:10.1080/02331934.2018.1518447