585 research outputs found
Boundaries of Amplituhedra and NMHV Symbol Alphabets at Two Loops
In this sequel to arXiv:1711.11507 we classify the boundaries of amplituhedra
relevant for determining the branch points of general two-loop amplitudes in
planar super-Yang-Mills theory. We explain the connection to
on-shell diagrams, which serves as a useful cross-check. We determine the
branch points of all two-loop NMHV amplitudes by solving the Landau equations
for the relevant configurations and are led thereby to a conjecture for the
symbol alphabets of all such amplitudes.Comment: 42 pages, 6 figures, 8 tables; v2: minor corrections and improvement
On the relationship between standard intersection cuts, lift-and-project cuts, and generalized intersection cuts
We examine the connections between the classes of cuts in the title. We show that lift-and-project (L&P) cuts from a given disjunction are equivalent to generalized intersection cuts from the family of polyhedra obtained by taking positive combinations of the complements of the inequalities of each term of the disjunction. While L&P cuts from split disjunctions are known to be equivalent to standard intersection cuts (SICs) from the strip obtained by complementing the terms of the split, we show that L&P cuts from more general disjunctions may not be equivalent to any SIC. In particular, we give easily verifiable necessary and sufficient conditions for a L&P cut from a given disjunction D to be equivalent to a SIC from the polyhedral counterpart of D. Irregular L&P cuts, i.e. those that violate these conditions, have interesting properties. For instance, unlike the regular ones, they may cut off part of the corner polyhedron associated with the LP solution from which they are derived. Furthermore, they are not exceptional: their frequency exceeds that of regular cuts. A numerical example illustrates some of the above properties. © 2016 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Societ
All-Helicity Symbol Alphabets from Unwound Amplituhedra
We review an algorithm for determining the branch points of general
amplitudes in planar super-Yang-Mills theory from amplituhedra.
We demonstrate how to use the recent reformulation of amplituhedra in terms of
`sign flips' in order to streamline the application of this algorithm to
amplitudes of any helicity. In this way we recover the known branch points of
all one-loop amplitudes, and we find an `emergent positivity' on boundaries of
amplituhedra.Comment: 38 pages, 5 figures, 1 big table; v2: minor corrections and
improvement
Branching via Cutting Plane Selection: Improving Hybrid Branching
Cutting planes and branching are two of the most important algorithms for
solving mixed-integer linear programs. For both algorithms, disjunctions play
an important role, being used both as branching candidates and as the
foundation for some cutting planes. We relate branching decisions and cutting
planes to each other through the underlying disjunctions that they are based
on, with a focus on Gomory mixed-integer cuts and their corresponding split
disjunctions. We show that selecting branching decisions based on quality
measures of Gomory mixed-integer cuts leads to relatively small
branch-and-bound trees, and that the result improves when using cuts that more
accurately represent the branching decisions. Finally, we show how the history
of previously computed Gomory mixed-integer cuts can be used to improve the
performance of the state-of-the-art hybrid branching rule of SCIP. Our results
show a 4\% decrease in solve time, and an 8\% decrease in number of nodes over
affected instances of MIPLIB 2017
Relaxations of mixed integer sets from lattice-free polyhedra
This paper gives an introduction to a recently established link between the geometry of numbers and mixed integer optimization. The main focus is to provide a review of families of lattice-free polyhedra and their use in a disjunctive programming approach. The use of lattice-free polyhedra in the context of deriving and explaining cutting planes for mixed integer programs is not only mathematically interesting, but it leads to some fundamental new discoveries, such as an understanding under which conditions cutting planes algorithms converge finitel
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