Norm Based Gap Function

Norm Based Gap Function implementation

Description

Reference documentation for a simple norm based gap function (where if the norm is set to 1, i.e. \(p=1\), we take the absolute difference).

This gap function utilises a norm based approach to computing the gap. It depends on the assignment method what variables are used to compute the gap. In general the user has to choose the norm (norm 1, norm 2, …, norm p) to apply and we also allow the user to choose whether or not the result is to be averaged or not, i.e., divide the result by the number of entries considered. It also hosts a stop criterion to determine when the gap is deemed small enough to be considered sufficient. This criterion is user configurable as well. Mathematically, the norm based gap is defined as follows:

\(\Large \delta = {1 \over |\mathbf{x}|} \sqrt[\leftroot{-3}\uproot{3}p]{ (x_1^{i-1}-x_1^i)^p + ... + (x_{|\mathbf{x}|}^{i-1}-x_{|\mathbf{x}|}^i)^p}\)

with

\(\delta = \textrm{gap},\\ p = \textrm{norm to apply, e.g. norm 2 requires } p=2,\\ i = \textrm{iteration},\\ \mathbf{x} = \textrm{vector of the variable }x \textrm{ we take the gap over between iterations, e.g., travel time cost},\\\)

When the averaged option is not activated, the portion of the equation where we multiply by \({1 \over |\mathbf{x}|}\) is left out

Defaults

Name Value Type Unit
Stop criterion Object instance Default Stop Criterion N/A
Norm (\(p\)) 1 Integer N/A
Averaged TRUE boolean N/A

Methods

This class exposes the following methods

Name Description
.set_norm(+) Override the default norm \(p\)
.set_averaged(+) Override the default averaged setting

Properties

This class exposes the following properties:

Property Availability Description
.stop_criterion Always Access to the stop criterion instance

Example 1

from planit import *

# choose to run PLANit with a the Static Link Transmission Model assignment method
# This has the norm based gap as a default already
planit_instance = Planit()
planit_instance.project.set(TrafficAssignment.SLTM)

# while not needed it can be set explicitly via
# planit_instance.project.assignment.set(GapFunction.NORM_BASED) 

# change the norm to use from 1 to 2
planit_instance.project.assignment.gap_function.set_norm(2)

See also

Traffic Assignment on what traffic assignment methods are available
StopCriterion default implementation

Source code

Class GapFunctionWrapper in projectwrappers.py

.stop_criterion

A Norm Based Gap Function method

set_averaged(+)

A Norm Based Relative Gap Function method

set_norm(+)

A Norm Based Relative Gap Function method

Last modified January 1, 0001