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Shapes and recession cones in mixed-integer convex representability

MATHEMATICAL programming 2024年第1-2期204卷 739-752页

作者： Zadik, Ilias Lubin, Miles Vielma, Juan Pablo MIT Dept Math Boston MA 02139 USA

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.

关键词： mixed-integer convex optimization mixed-integer programming formulations mixed-integer programming representability

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Bi-objective p-hub Location Problems

Bi-objective p-hub Location Problems

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作者： Fatih Bilen University of Waterloo

学位级别：硕士

In this thesis, we introduce, model, and solve bi-objective hub location problems. The two well-known hub location problems from the literature, the p-hub median and p-hub center problems, are unified under a bi-objective setting considering the single, multiple, and r-allocation strategies. We developed a 3-index and a 4-index mixed-integer program- ming formulation for each of the allocation strategies. All the formulations are tested on the CAB dataset from the literature using a commercial optimization software. We observe the effect of different priorities given to the objectives on the locations of hub nodes, allo- cations, and the CPU time requirements with different allocation strategies under different values of problem parameters.

关键词： p-hub median p-hub center hub location mixed-integer programming formulations

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