# A Roadmap for Optimization and Polyhedral Geometry softwares interactions

## (around Sage, (PyQ)Normaliz, SCIP, e-antic, Polymake, and others)

This page is intended to describe the current status of development of Sage related to optimization and polyhedral geometry.

## Future & Current Projects

- Extend the usage of Normaliz in Sage
- Extend the polyhedral geometry component of sage
- Polyhedral arrangements and complexes

## Meetings

Sage Days 84 on Polytopes in Olot

## Tickets

Here are some practical lists of tickets:

Tickets from the Sage Days 84 March 2017

Tickets from the IMA Coding Sprint - April 2018

Current tickets of the Geometry component

## Meta-tickets

There are already a few meta-tickets related to this topic:

Polytopes, lattice (integer) point counting / enumeration, and their applications

Improvements to MixedIntegerLinearProgram, its backends, and InteractiveLinearProgram

## To do

Test the algebraic polytopes #25097

Make Polyhedron_normaliz accept algebraic stuff, related to #25097 (adapt _init_from_Hrepresentation)

- Add known irrational polytope to the library of polytopes
repr of NumberFields (the parents) should indicate its embedding if there is one #21161 (this ticket will help in debugging and have good feedback)

- Thematic Tutorial for Py(Q)Normaliz installation/Usage in Sage
- Increase documentation in Py(Q)Normaliz
Deprecate the opacity argument for plot3d and use alpha #21984

Make Ehrhart (quasi)polynomial accessible through normaliz (and consisten with the current one) #18957

Adjust the new features of the normaliz backend to the latest version #25091 #25090

## Progress Report Timeline

### April 2018

#### Sage

Here are improvements realized during the Coding Sprints at the IMA in Minneapolis to objects related to polyhedron in Sage:

Thematic tutorial on polyhedron in Sage #22572

Make features of normaliz accessible #25091

`number_field_elements_from_algebraics`should create embedded number field elements #20181 Sage-devel discussionHelper functions and improvements in the normaliz backend #25097

Change error message in construction of polyhedron object #24835

Improve the output of repr_pretty for Polyhedron #24837

Implement stacking onto a face of a polyhedron #24847

Implement join of polytopes #24848

Implement one point suspension and face split of polyhedron #24886

The polar of a polyhedron should carry the backend used #25081

Construct RationalPolyhedralFan from possibly overlapping cones #25122

polygon3d ignores the "alpha" (and equivalent "opacity") argument #25095

Update normaliz 3.5.3 and pynormaliz 1.12 in Sage. Now Positive review. #22984

PolyhedralComplex: preliminary implementation, see #20877

#### (Q)Normaliz

http://normaliz.uni-osnabrueck.de

Upgrade normaliz to 3.5.3 and pynormaliz to 1.12 #22984: repair of performance loss

Upgrade normaliz to 3.5.4 and pynormaliz to 1.16 #25090: Ehrhart series and triangulation of polytope also for inhomogeneous input

Algebraic polytopes in Qnormaliz using e-antic #25097: convex hull computation completed, improvement of Qnormaliz input and output, control of interrupts and parallelization via libQnormaliz

fixed: getProjectionCone might segfault https://github.com/Normaliz/Normaliz/issues/180

fixed: BadInputException for ProjectCone https://github.com/Normaliz/Normaliz/issues/181

- General getters for cone properties. The getters can be used in future interfaces to make access to Normaliz cone properties easier
- resetGrading method for cone: Can be used to replace the grading and redo certain Normaliz computation without having to delete
- degree indepenent properties, such as hilbert bases.

- documentation of QNormaliz written
- extensive tests of QNormaliz
- link from libQnormaliz to libnormaliz in preparation for lattice point computation in algebraic polytopes
- further attempts to improve QNormaliz output (not yet complete)
- work on the C++ wrapper of e-antic
- Normaliz 3.5.4 released

#### Py(Q)Normaliz

https://pypi.python.org/pypi/PyNormaliz https://github.com/Normaliz/PyNormaliz

PyNormaliz (last version is 1.17):

- All computation goals and output properties from Normaliz 3.5.4 are supported
It is possible to give functions RationalHandler, VectorHandler, and MatrixHandler to NmzResult to construct rationals, vectors, and matrices

Some documentation in the sources of NormalizModule.cpp

- Functionality to copy cone
Functionality to reset grading of a cone in PyNormaliz

PyQNormaliz (current beta is 0.2, not available on PIPy):

- All QNormaliz compute targets can be accessed
- Input of number_field data as string
- Input of number_field elements as list of 2-tuples, or as string
Ouput handlers RationalHandler, NumberfieldElementHandler, VectorHandler, and MatrixHandler

See example worksheet: https://nbviewer.jupyter.org/github/sebasguts/PyQNormaliz/blob/master/examples/Dodecahedron.ipynb

Upgrade normaliz to 3.5.3 and pynormaliz to 1.12 #22984

Upgraded pynormaliz to 1.16 https://github.com/Normaliz/PyNormaliz

Started PyQNormaliz. https://github.com/sebasguts/PyQNormaliz

#### PyScipOpt

https://github.com/SCIP-Interfaces/PySCIPOpt

Sage devel discussion

#### Polymake

Upgrade polymake to version 3.2r2 (current interface breaks) #24905

### Sage 8.2

- Improvements:
- Library of polytopes accepts a backend parameter
- Fixed _facet_adjacency_matrix of non-full-dim. polyhedron
- Optimize count lattice points of polytopes

- New methods for polyhedron:
- get_integral_point

### Sage 8.1

- Improvements:
- Fixed equation handling in .to_linear_program and integral_points_count
- Fixed sage_input to include the backend used
- Improved the polyhedron constructor
- Fixed .affine_hull() raising an assertion error
- Volume function handling better the different measures depending on the engine used

### Sage 8.0

Here is a sample of changes related to Polyhedral Geometry in this release.

- Highlights:
- Integration with Latte (function .integrate() )
- New class for Voronoi Diagrams
- Backend polymake for polyhedron

- Improvements:
Fixed CartesianProduct with distinct dimensions

- Fixed an error in .is_lattice_polytope()
- Fixed overflow in integral_points()
- Fixed normaliz backend with empty polyhedron

- New methods for Polyhedron:
- integral_points_count()

### Sage 7.6

Here is a sample of changes related to Polyhedral Geometry in this release:

- Improvements:
- documentation of Combinatorial and Discrete Geometry
- combinatorial_automorphism_group
- Fixes on .volume()
- Fixes in integral_points()

- New methods for Polyhedron:
- .is_inscribed()
- .neighborliness()
- .is_neighborly()
- .normal_fan()
- .face_fan()
- .is_combinatorially_isomorphic()
- .face_truncation()

New methods for SimplicialComplexes:

- star of a face
- stellar subdivision

- Generic interface to latte integrale:
- count points

## A list of other Polyhedral Geometry softwares

What are other softwares dealing with polyhedral geometry? Here is a non-exhaustive list.

- Maple:
https://www.maplesoft.com/support/help/Maple/view.aspx?path=PolyhedralSets%2fPolyhedralSet (seems to be only rational and implemented in 2015).

http://qhull.org/ (seems old?)

https://www.maplesoft.com/products/maple/new_features/Maple2018/ComputationalGeometry.aspx

https://www.maplesoft.com/support/help/Maple/view.aspx?path=PolyhedralSets

- CGAL:
- A paper:
Jing & Maza https://dl.acm.org/citation.cfm?id=3177798

- Matlab:
- Matematica: