A python package for rounding polytopes.

# PolyRound

Efficient random sampling in convex polytopes relies on a 'rounding' preprocessing step, in which the polytope is rescaled so that the width is as uniform as possible across different dimensions. PolyRound rounds polytopes on the general form:

$P:=\{x \in \mathcal{R}^n: A_{eq} x = b_{eq}, A_{ineq} x \leq b_{ineq}\}$ with matrices $A_{eq} \in \mathcal{R}^{m,n}$ and $A_{ineq} \in \mathcal{R}^{k,n}$ and vectors $b_{eq} \in \mathcal{R}^{m}$ and $b_{ineq} \in \mathcal{R}^{k}$.

This formulation often arises in Systems Biology as the flux space of a metabolic network.

As output, PolyRound produces a polytope on the form $P^{r}:=\{v \in \mathcal{R}^l: A^{r}_{ineq}v \leq b^{r}_{ineq}\}$ where $l \leq n$ and the zero vector is a stricly interior point. For transforming points back to the original space, it also provides a matrix $S \in \mathcal{R}^{n,l}$ and a vector $t \in \mathcal{R}^{n}$, so that $x=Sv + t$.

Currently, PolyRound is supported for python 3.7 to 3.9.

PolyRound no longer depends on a Gurobi installation and uses optlang (https://github.com/opencobra/optlang) to delegate linear programs to GLPK in case Gurobi is not installed. However, PolyRound is more reliable with Gurobi. Free Gurobi licenses for academic use can be obtained at https://www.gurobi.com/. Once the license is installed, gurobipy can be installed directly through pip, or by getting the optional requirements by 'pip install -r optional_requirements.txt'.

An easy example of how to get started is presented in the jupyter notebook cells below.

They show how to:

1. create a polytope object from a file path
2. simplify, reduce, and round a polytope in separate steps, togehter with some printed checks
3. simplify, reduce and round a polytope in one step
4. save the rounded polytope
import os
from PolyRound.api import PolyRoundApi
from PolyRound.static_classes.lp_utils import ChebyshevFinder
from PolyRound.settings import PolyRoundSettings
from pathlib import Path

model_path = os.path.join("PolyRound", "models", "e_coli_core.xml")

# Create a settings object with the default settings.
settings = PolyRoundSettings()

# Import model and create Polytope object
polytope = PolyRoundApi.sbml_to_polytope(model_path)

# Remove redundant constraints and refunction inequality constraints that are de-facto equalities.
# Due to these inequalities, the polytope is empty (distance from chebyshev center to boundary is zero)
x, dist = ChebyshevFinder.chebyshev_center(polytope, settings)
print(dist)
simplified_polytope = PolyRoundApi.simplify_polytope(polytope)
# The simplified polytope has non-zero border distance
x, dist = ChebyshevFinder.chebyshev_center(simplified_polytope, settings)
print(dist)

# Embed the polytope in a space where it has non-zero volume
transformed_polytope = PolyRoundApi.transform_polytope(simplified_polytope)
# The distance from the chebyshev center to the boundary changes in the new coordinate system
x, dist = ChebyshevFinder.chebyshev_center(transformed_polytope, settings)
print(dist)

# Round the polytope
rounded_polytope = PolyRoundApi.round_polytope(transformed_polytope)
# After rounding, the distance from the chebyshev center to the boundary is set to be close to 1
x, dist = ChebyshevFinder.chebyshev_center(rounded_polytope, settings)
print(dist)

# The chebyshev center can be back transformed into an interior point in the simplified space.
print(simplified_polytope.border_distance(rounded_polytope.back_transform(x)))

# simplify, transform and round in one call
one_step_rounded_polytope = PolyRoundApi.simplify_transform_and_round(polytope)

# save to csv
out_csv_dir = os.path.join("PolyRound", "output", "e_coli_core")
Path(out_csv_dir).mkdir(parents=True, exist_ok=True)
PolyRoundApi.polytope_to_csvs(one_step_rounded_polytope, out_csv_dir)

# Special use case: remove redundant constraints without removing zero facettes. This will leave th polytope with its original border distance.
x, dist = ChebyshevFinder.chebyshev_center(polytope, settings)
print(dist)
settings.simplify_only = True
simplified_polytope = PolyRoundApi.simplify_polytope(polytope, settings=settings)
# The simplified polytope still has zero border distance
x, dist = ChebyshevFinder.chebyshev_center(simplified_polytope, settings)
print(dist)


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