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Mixed-Integer Cuts from Cyclic Groups

Matteo Fischetti, Cristiano Saturni
2006 Mathematical programming
Saturni, Mixed-Integer Cuts from Cyclic Groups  ...  of the following master cyclic group polyhedron: T (k, r) = conv{t ∈ Z k−1 + : k−1 i=1 (i/k) · t i ≡ r/k (mod 1)} (4) where k ≥ 2 (group order) and r ∈ {1, · · · , k − 1} are given integers • The space  ...

Mixed-Integer Cuts from Cyclic Groups [chapter]

Matteo Fischetti, Cristiano Saturni
2005 Lecture Notes in Computer Science
of the following master cyclic group polyhedron: T (k, r) = conv{t ∈ Z k−1 + : k−1 i=1 (i/k) · t i ≡ r/k (mod 1)} (4) where k ≥ 2 (group order) and r ∈ {1, · · · , k − 1} are given integers • The space  ...  separation problem is embedded into the original ILP model • Letchford and Lodi (2002) and Cornuejols, Li and Vandenbussche (2003) address specific subfamilies of cyclic-group cuts • To our knowledge  ...

On a generalization of the master cyclic group polyhedron

Sanjeeb Dash, Ricardo Fukasawa, Oktay Günlük
2008 Mathematical programming
Finally, we study the mixed-integer extension of the MEP and present an interpolation theorem that produces valid inequalities for general Mixed Integer Programming Problems using facets of the MEP.  ...  We study the Master Equality Polyhedron (MEP) which generalizes the Master Cyclic Group Polyhedron and the Master Knapsack Polyhedron.  ...  The Gomory mixed-integer cut (also known as the mixed-integer rounding (MIR) inequality) can be derived from a facet of P (n, r) [10] .  ...

Cyclic group and knapsack facets

Juli�n Ar�oz, Lisa Evans, Ralph E. Gomory, Ellis L. Johnson
2003 Mathematical programming
Any integer program may be relaxed to a group problem.  ...  We define the master cyclic group problem and several master knapsack problems, show the relationship between the problems, and give several classes of facet-defining inequalities for each problem, as  ...  For example, for n ≤ 7, every cyclic group facet, except one, is either a mixed integer cut itself or comes from one or more mappings of a mixed integer cut.  ...

On the strength of Gomory mixed-integer cuts as group cuts

Sanjeeb Dash, Oktay Günlük
2007 Mathematical programming
Further, it is well-known that GMI cuts can be derived from facets of Gomory's master cyclic group polyhedron and its mixed-integer extension studied by Gomory and Johnson.  ...  Gomory mixed-integer (GMI) cuts generated from optimal simplex tableaus are known to be useful in solving mixed-integer programs.  ...  GMI cuts for mixed-integer programs can similarly be derived from the mixed-integer extension of the master cyclic group polyhedron; see Gomory and Johnson [18] .  ...

Page 4241 of Mathematical Reviews Vol. , Issue 2004e [page]

2004 Mathematical Reviews
The paper describes a theory that allows one to automatically derive many cutting planes for integer or mixed integer programs from rows of a tableau belonging to a basic feasible solution of the LP relaxation  ...  From any facet of the master polyhedron one can derive a valid inequality in the non-basic variables cf the original (mixed) integer program that cuts off the present basic feasible solution.  ...

Valid inequalities based on simple mixed-integer sets

Sanjeeb Dash, Oktay Günlük
2005 Mathematical programming
Facets of this polyhedron give strong valid inequalities for general mixed-integer sets, such as the well-known Gomory mixed-integer cut.  ...  These inequalities also define facets of the master cyclic group polyhedron of Gomory.  ...  The well-known Gomory mixed-integer cut (GMIC) can be derived from a facet of P (n, r) [10] .  ...

Valid Inequalities Based on Simple Mixed-Integer Sets [chapter]

Sanjeeb Dash, Oktay Günlük
2004 Lecture Notes in Computer Science
Facets of this polyhedron give strong valid inequalities for general mixed-integer sets, such as the well-known Gomory mixed-integer cut.  ...  These inequalities also define facets of the master cyclic group polyhedron of Gomory.  ...  The well-known Gomory mixed-integer cut (GMIC) can be derived from a facet of P (n, r) [10] .  ...

Profit-based FMS dynamic part type selection over time for mid-term production planning

Kathryn E. Stecke, Eugeniusz Toczyłowski
1992 European Journal of Operational Research
At the upper level, a large-scale linear programming Master Problem is solved, the columns of which are generated by solving a nonlinear, mixed-integer, profit-based part type selection subproblem that  ...  Each cyclic schedule allows a subset of parts of different types to be released periodically in appropriate production ratios.  ...  Acknowledgments Kathy Stecke would like to acknowledge a Summer Grant from the Graduate School of Business at The University of Michigan and Eugeniusz Toczytowski acknowledges a Grant from the Polish Science  ...

Throughput-Optimal Sequences for Cyclically Operated Plants

Eckart Mayer, Utz-Uwe Haus, Jörg Raisch, Robert Weismantel
2008 Discrete event dynamic systems
The resulting scheduling problem is solved by deriving a mixed integer optimization problem from a discrete event model.  ...  In cyclic operation, a large number of entities is processed in an identical time scheme.  ...  Acknowledgement The authors gratefully acknowledge funding by "Deutsche Forschungsgemeinschaft (DFG)" via the DFG-Forschergruppe 468 "Methods from Discrete Mathematics for the Synthesis and Control of  ...

Page 1636 of Mathematical Reviews Vol. , Issue 88c [page]

1988 Mathematical Reviews
The mixed integer cyclic group problem is then considered and a dual problem given.  ...  A generalized method of marks for a problem of integer linear programming. Soviet J. Comput. Systems Sci. 24 (1986), no. 5, 90-97 (1987); translated from Izv. Akad. Nauk SSSR Tekhn.  ...

New inequalities for finite and infinite group problems from approximate lifting

Lisa A. Miller, Yanjun Li, Jean-Philippe P. Richard
2008 Naval Research Logistics
mixed-integer programming problems.  ...  These new inequalities not only illustrate the diversity of strong inequalities for the finite and infinite group problems, but also provide a large variety of new cutting planes for solving integer and  ...  These inequalities can then be used as cutting planes for general integer and mixed-integer programs. We call these cutting planes group cuts.  ...

Mixed-Integer Programming for Cycle Detection in Non-reversible Markov Processes [article]

Isabel Beckenbach, Leon Eifler, Konstantin Fackeldey, Ambros Gleixner, Andreas Grever, Marcus Weber, Jakob Witzig
2016 arXiv   pre-print
We address this difficulty using a mixed-integer programming model that allows us to compute a cycle of clusters with maximum net flow, i.e., large forward and small backward probability.  ...  In this paper, we present a new, optimization-based method to exhibit cyclic behavior in non-reversible stochastic processes.  ...  Mixed-Integer Programming for Cycle Detection in Non-reversible Markov Processes * Introduction Simulation data stemming from chemical or biological processes typically lead to a huge amount of data  ...

Mixed-Integer Programming for Cycle Detection in Nonreversible Markov Processes

Jakob Witzig, Isabel Beckenbach, Leon Eifler, Konstantin Fackeldey, Ambros Gleixner, Andreas Grever, Marcus Weber
2018 Multiscale Modeling & simulation
We address this difficulty using a mixed-integer programming model that allows us to compute a cycle of clusters with maximum net flow, i.e., large forward and small backward probability.  ...  In this paper, we present a new, optimization-based method to exhibit cyclic behavior in non-reversible stochastic processes.  ...  Mixed-Integer Programming A mixed-integer program (MIP) is an optimization problem that can be written in the form (P ) z M IP = min{c T x | Ax ≥ b, ≤ x ≤ u, x ∈ Z l × R n−l }, with objective function  ...

Page 2420 of Mathematical Reviews Vol. , Issue 92e [page]

1992 Mathematical Reviews
A completion of a cyclically ordered group was constructed by the author and J. Jakubik by means of cuts; they proved that each cyclically ordered group has a unique completion [Czechoslovak Math.  ...  A mixed product decomposition of a di- rected group is an isomorphism from the group to such a mixed product, and the system of all mixed product decompositions, modulo a natural equivalence relation,  ...
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