Shapes and recession cones in mixed-integer convex representability

Mixed-integer convex representable (MICP-R) sets are those sets that can be represented exactly through a mixed-integer convex programming formulation. Following up on recent work by Lubin et al. (in: Eisenbrand (ed) Integer Programming and Combinatorial Optimization - 19th International Conference,...

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Vydáno v:Mathematical programming Ročník 204; číslo 1-2; s. 739 - 752
Hlavní autoři: Zadik, Ilias, Lubin, Miles, Vielma, Juan Pablo
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2024
Témata:
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Short Communication
Theoretical
Mixed-integer programming formulations
Mixed-integer programming representability
90C25
Mixed-integer convex optimization
90C11
ISSN:0025-5610, 1436-4646
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Popis
Shrnutí:Mixed-integer convex representable (MICP-R) sets are those sets that can be represented exactly through a mixed-integer convex programming formulation. Following up on recent work by Lubin et al. (in: Eisenbrand (ed) Integer Programming and Combinatorial Optimization - 19th International Conference, Springer, Waterloo), (Math. Oper. Res. 47:720-749, 2022) we investigate structural geometric properties of MICP-R sets, which strongly differentiate them from the class of mixed-integer linear representable (MILP-R) sets. First, we provide an example of an MICP-R set which is the countably infinite union of convex sets with countably infinitely many different recession cones. This is in sharp contrast with MILP-R sets which are (countable) unions of polyhedra that share the same recession cone. Second, we provide an example of an MICP-R set which is the countably infinite union of polytopes all of which have different shapes (no pair is combinatorially equivalent, which implies they are not affine transformations of each other). Again, this is in sharp contrast with MILP-R sets which are (countable) unions of polyhedra that are all translations of a finite subset of themselves.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-023-01946-4