src.insertion_step

Module Contents

Functions

insertion_step(current_measure)

Insertion step & optimization step executed on a target measure.

multistart_descent(current_measure)

Uses multistart descent to search for the global minimizing curve.

is_close_to_stationaries(new_curve, new_curve_energy, stationary_curves, energy_curves) → bool

Checks if a given curve is close to the set of found stationary curves.

gradient_descent(curve, w_t, max_iter=None, init_step=None, limit_stepsize=None)

Applies the gradient descent to an input curve.
src.insertion_step.insertion_step(current_measure)

Insertion step & optimization step executed on a target measure.

Parameters
current_measuresrc.classes.measure

Target measure to apply the inserion + optimization step

Returns
new_measuresrc.classes.measure
exit_flagint

0 if no new inserted curve was found. 1 else.

src.insertion_step.multistart_descent(current_measure)

Uses multistart descent to search for the global minimizing curve.

The multistart method corresponds to descent multiple randomly generated curves and to record the resulting stationary point of this descent expecting to find with this method the global minimizing curve. Some details:

Parameters
current_measuresrc.classes.measure

the current iterate of the algorithm.

Returns
stationary_curveslist[src.classes.curve]

list of the found stationary points of the insertion step problem.

energy_curvesnumpy.ndarray

respective energy of the found stationary_curves, sorted in ascending order.

Notes

  • To decrease the number of descents, this method routinely checks

if the current descended curve is close to the already known ones. If so, it stops and discards the curve. - The descented curves are proposed by src.insertion_mod.propose() It consists of: already known curves, crossover curves, random ones. - If a crossover curve gets too close to a stationary curve earlier than the first check, it is not counted as an attempt.

src.insertion_step.is_close_to_stationaries(new_curve, new_curve_energy, stationary_curves, energy_curves) → bool

Checks if a given curve is close to the set of found stationary curves.

The distance is measured with the H^1 norm, and the threshold is set by config.multistart_taboo_dist.

Parameters
new_curvesrc.classes.curve

Curve to check if it is close to the stationary set

new_curve_energyfloat

Energy of the curve to check

stationary_curveslist[src.classes.curve]

List of found stationary curves

energy_curvesnumpy.ndarray

Energies of the found stationary curves sorted in ascendent order.

Notes

The energy_curves are used to accelerate the comparisons. To avoid with the whole set of found stationary curves.

src.insertion_step.gradient_descent(curve, w_t, max_iter=None, init_step=None, limit_stepsize=None)

Applies the gradient descent to an input curve.

The function to minimize F(γ) is defined via the dual variable. The Applied gradient descent is the Armijo with backtracking, with stopping condition reached when the stepsize reaches a predefined value.

Parameters
curvesrc.classes.curve

Curve to be descended.

w_tsrc.classes.dual_variable

Dual variable associated to the current iterate.

max_iterint, optional

A bound on the number of iterations. Defaults to config.multistart_descent_max_iter.

init_stepfloat, optional

Defines the initial step of the descent method. Defaults to config.multistart_descent_init_step.

limit_stepsizefloat, optional

The stopping condition for the gradient descent. Defaults to config.multistart_descent_limit_stepsize

Returns
src.classes.curve

Notes

As described in the paper, the gradient descent assumes that the input curve has negative energy: F(γ) < 0.