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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| Vydané v: | Optimization Ročník 68; číslo 10; s. 2039 - 2054 |
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| Hlavní autori: | , , |
| Médium: | Journal Article |
| Jazyk: | English |
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Philadelphia
Taylor & Francis
03.10.2019
Taylor & Francis LLC |
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| ISSN: | 0233-1934, 1029-4945 |
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| Abstract | 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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| AbstractList | 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. |
| Author | Löhne, Andreas Ciripoi, Daniel Weißing, Benjamin |
| Author_xml | – sequence: 1 givenname: Daniel surname: Ciripoi fullname: Ciripoi, Daniel organization: Department of Mathematics, Friedrich Schiller University Jena – sequence: 2 givenname: Andreas surname: Löhne fullname: Löhne, Andreas email: andreas.loehne@uni-jena.de organization: Department of Mathematics, Friedrich Schiller University Jena – sequence: 3 givenname: Benjamin surname: Weißing fullname: Weißing, Benjamin organization: Department of Mathematics, Friedrich Schiller University Jena |
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| Cites_doi | 10.1142/8527 10.1007/978-0-387-68407-9 10.1007/978-3-642-18351-5 10.1007/s00186-016-0554-0 10.1007/s10898-013-0098-2 10.1137/060674831 10.1023/A:1008215702611 |
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| SubjectTerms | 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 |
| Title | Calculus of convex polyhedra and polyhedral convex functions by utilizing a multiple objective linear programming solver |
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