variable neighbourhood search...

Variable Neighbourhood Search (VNS)

Key Idea: systematically change neighbourhoods during search

Motivation:

I recall: changing neighbourhoods can help escape local optima

I a global optimum is locally optimal w.r.t. all neighbourhoodstructures

I principle of VNS: change the neighbourhood during the search

Heuristic Optimization 2015 129

I main VNS variants

I variable neighbourhood descent (VND, already discussed)

I basic variable neighborhood search

I reduced variable neighborhood search

I variable neighborhood decomposition search


How to generate the various neighborhood structures?

I for many problems di↵erent neighborhood structures (localsearches) exist / are in use

I use k-exchange neighborhoods; these can be naturallyextended

I many neighborhood structures are associated with distancemeasures: define neighbourhoods in dependence of thedistances between solutions


basic VNS

I uses neighborhood structures Nk

, k = 1, . . . , kmax

I iterative improvement in N1

I other neighborhoods are explored only randomly

I exploration in other neighborhoods are perturbations in theILS sense

I perturbation is systematically varied

I acceptance criterion Better(s⇤, s⇤0)


Basic VNS — Procedural view

procedure basic VNSs0 GenerateInitialSolution, choose {N

k

}, k = 1, . . . , kmax

repeats 0 RandomSolution(N

k

(s⇤))s⇤0 LocalSearch(s 0) % local search w.r.t. N1

if f (s⇤0) < f (s⇤) thens⇤ s⇤0

k 1else

k k + 1until termination condition

end


Basic VNS — variants

I order of the neighborhoods

I forward VNS: start with k = 1 and increase k by one if nobetter solution is found; otherwise set k 1

I backward VNS: start with k = kmax

and decrease k by one ifno better solution is found

I extended version: parameters kmin

and kstep

; set k kmin

andincrease k by k

step

if no better solution is found

I acceptance of worse solutions

I Skewed VNS: accept if

f (s⇤0)� ↵d(s⇤, s⇤0) < f (s⇤)

d(s⇤, s⇤0) measures distance between candidate solutions


Reduced VNS

I same as basic VNS except that no iterative improvementprocedure is applied

I only explores randomly di↵erent neighborhoods

I goal: reach quickly good quality solutions for large instances


Variable Neighborhood Decomposition Search

I central idea

I generate subproblems by keeping all but k solutioncomponents fixed

I apply local search only to the k “free” components

1

2

3

45

6

7

8

fix red edges and

define subproblem

1

2 3

4

5

67

8

I related approaches: POPMUSIC, MIMAUSA, etc.


VNDS — Procedural view

procedure VNDSs0 GenerateInitialSolution, choose {N

k

}, k = 1, . . . , kmax

repeats 0 RandomSolution(N

k

(s))t FixComponents(s 0, s)t⇤ LocalSearch(t) % local search w.r.t. N1

s 00 InjectComponents(t⇤, s 0)if f (s 00) < f (s) then

s s 00

k 1else

k k + 1until termination condition

end


relationship between ILS and VNS

I the two SLS methods are based on di↵erent underlying“philosophies”

I they are similar in many respects

I ILS apears to be in literature more flexible w.r.t. optimizationof the interaction of modules

I VNS gives place to approaches like VND for obtaining morepowerful local search approaches


Greedy Randomised Adaptive Search Procedures

Key Idea: Combine randomised constructive search withsubsequent perturbative local search.

Motivation:

I Candidate solutions obtained from construction heuristics canoften be substantially improved by perturbative local search.

I Perturbative local search methods typically often requiresubstantially fewer steps to reach high-quality solutionswhen initialised using greedy constructive search rather thanrandom picking.

I By iterating cycles of constructive + perturbative search,further performance improvements can be achieved.


Greedy Randomised “Adaptive” Search Procedure (GRASP):

While termination criterion is not satisfied:|| generate candidate solution s using|| subsidiary greedy randomised constructive search||b perform subsidiary local search on s

Note:

Randomisation in constructive search ensures that a large numberof good starting points for subsidiary local search is obtained.


Restricted candidate lists (RCLs)

I Each step of constructive search adds a solution componentselected uniformly at random from a restricted candidate list(RCL).

I RCLs are constructed in each step using a heuristic function h.

I RCLs based on cardinality restriction comprise the kbest-ranked solution components. (k is a parameterof the algorithm.)

I RCLs based on value restriction comprise all solutioncomponents l for which h(l) h

min

+ ↵ · (hmax

� hmin

),where h

min

= minimal value of h and hmax

= maximal valueof h for any l . (↵ is a parameter of the algorithm.)


Note:

I Constructive search in GRASP is ‘adaptive’:Heuristic value of solution component to be added togiven partial candidate solution r may depend onsolution components present in r .

I Variants of GRASP without perturbative local search phase(aka semi-greedy heuristics) typically do not reachthe performance of GRASP with perturbative local search.


Example: GRASP for SAT [Resende and Feo, 1996]

I Given: CNF formula F over variables x1, . . . , xn

I Subsidiary constructive search:

I start from empty variable assignment

I in each step, add one atomic assignment (i.e., assignment ofa truth value to a currently unassigned variable)

I heuristic function h(i , v) := number of clauses thatbecome satisfied as a consequence of assigning x

i

:= v

I RCLs based on cardinality restriction (contain fixed number kof atomic assignments with largest heuristic values)

I Subsidiary local search:

I iterative best improvement using 1-flip neighbourhood

I terminates when model has been found or given number ofsteps has been exceeded


GRASP has been applied to many combinatorial problems,including:

I SAT, MAX-SAT

I the Quadratic Assignment Problem

I various scheduling problems

Extensions and improvements of GRASP:

I reactive GRASP (e.g., dynamic adaptation of ↵during search)

I combinations of GRASP with tabu search, path relinking andother SLS methods


Iterated Greedy

Key Idea: iterate over greedy construction heuristics throughdestruction and construction phases

Motivation:

I start solution construction from partial solutions to avoidreconstruction from scratch

I keep features of the best solutions to improve solution quality

I if few construction steps are to be executed, greedy heuristicsare fast

I adding a subsidiary local search phase may further improveperformance


Iterated Greedy (IG):

While termination criterion is not satisfied:|| generate candidate solution s using|| subsidiary greedy constructive searchWhile termination criterion is not satisfied:|| r := s|| apply solution destruction on s|| perform subsidiary greedy constructive search on s|||||| based on acceptance criterion,b keep s or revert to s := r

Note:

I subsidiary local search after solution reconstruction cansubstantially improve performance


Iterated Greedy (IG):

While termination criterion is not satisfied:|| generate candidate solution s using|| subsidiary greedy constructive searchWhile termination criterion is not satisfied:|| r := s|| apply solution destruction on s|| perform subsidiary greedy constructive search on s|| perform subsidiary local search on s|| based on acceptance criterion,b keep s or revert to s := r

Note:

I subsidiary local search after solution reconstruction cansubstantially improve performance


IG—main issues

I destruction phase

I fixed vs. variable size of destructionI stochastic vs. deterministic destructionI uniform vs. biased destruction

I construction phase

I not every construction heuristic is necessarily usefulI typically, adaptive construction heuristics preferableI speed of the construction heuristic is an issue

I acceptance criterion

I very much the same issue as in ILS


IG — enhancements

I usage of history information to bias destructive/constructivephase

I combination with local search in the constructive phase

I use local search to improve full solutions destruction / construction phases can be seen as aperturbation mechanism in ILS

I usage of elements of tree search algorithms such as lowerbounds on the completion of a solution or constraintpropagation techniques


Example: IG for SCP [Jacobs, Brusco, 1995]

I Given:I finite set A = {a1, . . . , am} of objects

I family B = {B1, . . .Bn

} of subsets of A that covers AI weight function w : B 7! R+

I C ✓ B covers A if every element in A appears in at least oneset in C, i.e. if

SC = A

I Goal:

I find a subset C⇤ ✓ B of minimum total weight that covers A.


Example: IG for SCP, continued ..

I assumption: all subsets from B are ordered according tonondecreasing costs

I construct initial solution using a greedy heuristic based on twosteps

I randomly select an uncovered object ai

I add the lowest cost subset that covers ai

I the destruction phase removes a fixed number of k1|C|subsets; k1 is a parameter


I the construction phase proceeds as

I build a candidate set containing subsets with cost of less thank2 · f (C)

I compute cover value �j

= wj

/dj

dj

: number of additional objects covered by adding subset bj

I add a subset with minimum cover value

I complete solution is post-processed by removing redundantsubsets

I acceptance criterion: Metropolis condition from PII

I computational experience

I good performance with this simple approach

I more recent IG variants are state-of-the-art algorithms for SCP


I IG has been re-invented several times; names include

I simulated annealing, ruin–and–recreate, iterative flattening,iterative construction search, large neighborhood search, ..

I close relationship to iterative improvement in largeneighbourhoods

I analogous extension to greedy heuristics as ILS to local search

I for some applications so far excellent results

I can give lead to e↵ective combinations of tree search andlocal search heuristics


Population-based SLS Methods

SLS methods discussed so far manipulate one candidate solution ofgiven problem instance in each search step.

Straightforward extension: Use population (i.e., set) ofcandidate solutions instead.

Note:

I The use of populations provides a generic way to achievesearch diversification.

I Population-based SLS methods fit into the general definitionfrom Chapter 1 by treating sets of candidate solutions assearch positions.


Ant Colony Optimisation (1)

Key idea: Can be seen as population-based constructive approachwhere a population of agents – (artificial) ants – communicate viacommon memory – (simulated) pheromone trails.

Inspired by foraging behaviour of real ants:

I Ants often communicate via chemicals known as pheromones,which are deposited on the ground in the form of trails.(This is a form of stigmergy: indirect communication viamanipulation of a common environment.)

I Pheromone trails provide the basis for (stochastic)trail-following behaviour underlying, e.g., the collectiveability to find shortest paths between a food source andthe nest.


Double bridge experiments Deneubourg++

I laboratory colonies of Iridomyrmex humilis

I ants deposit pheromone while walking from food sources tonest and vice versa

I ants tend to choose, in probability, paths marked by strongpheromone concentrations

Nest Food600

15 cm

Nest Food1 2


I equal length bridges: convergence to a single path

I di↵erent length paths: convergence to short path

0

5 0

100

0-20 20-40 40-60 60-80 80-100

% of traffic on the short branch

% o

f e

xp

eri

me

nts

0

50

100

0-20 20-40 40-60 60-80 80-100

% of traffic on one of the branches

% o

f exp

erim

ents


I a stochastic model was derived from the experiments andverified in simulations

I functional form of transition probability

pi ,a =

(k + ⌧i ,a)↵

(k + ⌧i ,a)↵ + (k + ⌧

i ,a0)↵

I pi ,a: probability of choosing branch a when being at decision

point i⌧i ,a: corresponding pheromone concentration

I good fit to experimental data with ↵ = 2


Towards artificial ants

I real ant colonies are solving shortest path problems

I ACO takes elements from real ant behavior to solve morecomplex problems than real ants

I In ACO, artificial ants are stochastic solution constructionprocedures that probabilistically build solutions exploiting

I (artificial) pheromone trails that change at run time to reflectthe agents’ acquired search experience

I heuristic information on the problem instance being solved


Ant Colony Optimisation

Application to combinatorial problems:[Dorigo et al. 1991, 1996]

I Ants iteratively construct candidate solutions.

I Solution construction is probabilistically biased bypheromone trail information, heuristic information andpartial candidate solution of each ant.

I Pheromone trails are modified during the search processto reflect collective experience.


Ant Colony Optimisation (ACO):

initialise pheromone trails

While termination criterion is not satisfied:|| generate population sp of candidate solutions|| using subsidiary randomised constructive search|||| perform subsidiary local search on sp||b update pheromone trails based on sp


Note:

I In each cycle, each ant creates one candidate solutionusing a constructive search procedure.

I Subsidiary local search is applied to individual candidatesolutions.

I All pheromone trails are initialised to the same value, ⌧0.

I Pheromone update typically comprises uniform decrease ofall trail levels (evaporation) and increase of some trail levelsbased on candidate solutions obtained from construction +local search.

I Termination criterion can include conditions on make-up ofcurrent population, e.g., variation in solution quality ordistance between individual candidate solutions.


i

j

k

g

! ij " ij

?,


Example: A simple ACO algorithm for the TSP (1)

(Variant of Ant System for the TSP [Dorigo et al., 1991; 1996].)

I Search space and solution set as usual (all Hamiltonian cyclesin given graph G ).

I Associate pheromone trails ⌧ij

with each edge (i , j) in G .

I Use heuristic values ⌘ij

:= 1/w((i , j)).

I Initialise all weights to a small value ⌧0 (parameter).

I Constructive search: Each ant starts with randomly chosenvertex and iteratively extends partial round trip � by selectingvertex not contained in � with probability

[⌧ij

]↵ · [⌘ij

]�Pl2N0(i)[⌧il ]

↵ · [⌘ij

]�Heuristic Optimization 2015 164


I Subsidiary local search: Perform iterative improvementbased on standard 2-exchange neighbourhood on eachcandidate solution in population (until local minimum isreached).

I Update pheromone trail levels according to

⌧ij

:= (1� ⇢) · ⌧ij

+X

s

02sp0�(i , j , s 0)

where �(i , j , s 0) := 1/f (s 0) if edge (i , j) is contained inthe cycle represented by s 0, and 0 otherwise.

Motivation: Edges belonging to highest-quality candidatesolutions and/or that have been used by many ants should bepreferably used in subsequent constructions.



I Termination: After fixed number of iterations(= construction + local search phases).

Note:

I Ants can be seen as walking along edges of given graph(using memory to ensure their tours correspond toHamiltonian cycles) and depositing pheromone to reinforceedges of tours.

I Original Ant System did not include subsidiary local searchprocedure (leading to worse performance compared tothe algorithm presented here)


ACO algorithm Authors Year TSP

Ant System Dorigo, Maniezzo, Colorni 1991 yesElitist AS Dorigo 1992 yesAnt-Q Gambardella & Dorigo 1995 yesAnt Colony System Dorigo & Gambardella 1996 yesMMAS Stutzle & Hoos 1996 yesRank-based AS Bullnheimer, Hartl, Strauss 1997 yesANTS Maniezzo 1998 noBest-Worst AS Cordon, et al. 2000 yesHyper-cube ACO Blum, Roli, Dorigo 2001 noPopulation-based ACO Guntsch, Middendorf 2002 yesBeam-ACO Blum 2004 no


MAX–MIN Ant System

I extension of Ant System with stronger exploitation of bestsolutions and additional mechanism to avoid search stagnation

I exploitation: only the iteration-best or best-so-far ant depositpheromone

⌧ij

(t + 1) = (1� ⇢) · ⌧ij

(t) +�⌧bestij

I frequently, a schedule for choosing between iteration-best andbest-so-far update is used


I stagnation avoidance: additional limits on the feasiblepheromone trails

I for all ⌧ij

(t) we have: ⌧min

⌧ij

(t) ⌧max

I counteracts stagnation of search through aggressivepheromone update

I heuristics for determining ⌧min

and ⌧max

I stagnation avoidance 2: occasional pheromone trailre-initialization when MMAS has converged

I increase of exploration: pheromone values are initialized to⌧max

to have less pronounced di↵erences in selectionprobabilities


Ant Colony System (ACS) Gambardella, Dorigo 1996, 1997

I pseudo-random proportional action choice rule

I with probability q0 an ant k located at city i chooses successorcity j with maximal ⌧

ij

(t) · [⌘ij

]� (exploitation)

I with probability 1� q0 an ant k chooses the successor city jaccording to action choice rule used in Ant System (biasedexploration)


I global pheromone update rule (exploitation)

I best-so-far solution T gb deposits pheromone after eachiteration

⌧ij

(t) = (1� ⇢) · ⌧ij

(t � 1) + ⇢ ·�⌧ gbij

(t � 1),

where �⌧ gbij

(t) = 1/Lgb.

I pheromone update only a↵ects edges in T gb!

I limited tests with iteration-best solution; however, best-so-farupdate is recommended


I local pheromone update rule

I ants “eat away” pheromone on the edges just crossed

⌧ij

= (1� ⇠) · ⌧ij

+ ⇠ · ⌧0I ⌧0 is some small constant value

I increases exploration by reducing the desirability of frequentlyused edges

I in Ant-Q (predecessor of ACS): ⌧0 = � ·maxl2N k

j

{⌧jl

}


solution quality versus time

d198 rat783

0

2

4

6

8

10

12

14

16

18

20

22

0.1 1 10 100

pe

rce

nta

ge

devia

tio

n f

rom

optim

um

CPU time [sec]

ASEASRAS

MMASACS

0

5

10

15

20

25

30

35

10 100 1000 10000

percentage deviation from optimum

CPU time [sec]

ASEASRAS

MMASACS

(typical parameter settings for high final solution quality)


behavior of ACO algorithms

average �-branching factor average distance among tours

1

2

3

4

5

6

7

8

1 10 100 1000 10000

ave

rag

e b

ran

chin

g fa

cto

r

no. iterations

ASEASRAS

MMASACS

10

20

30

40

50

60

70

80

1 10 100 1000 10000

ave

rag

e d

ista

nce

am

on

g t

ou

rs

no. iterations

ASEASRAS

MMASACS

(typical parameter settings for high final solution quality)


Enhancements:

I use of look-ahead in construction phase;

I start of solution construction from partial solutions(memory-based schemes, ideas gleaned from iterated greedy);

I combination of ants with techniques from tree search such as

I lower bounding information

I combination with beam search

I constraint programming (constraint propagation)


Applying ACO

copyright Christian Blum


Ant Colony Optimisation . . .

I has been applied very successfully to a wide range ofcombinatorial problems, including

I various scheduling problems,

I various vehicle routing problems, and

I various bio-informatics problems such as protein–liganddocking

I underlies new high-performance algorithms for dynamicoptimisation problems, such as routing in telecommunicationsnetworks [Di Caro and Dorigo, 1998].


Note:

A general algorithmic framework for solving static and dynamiccombinatorial problems using ACO techniques is provided by theACO metaheuristic [Dorigo and Di Caro, 1999; Dorigo et al., 1999].

For further details on Ant Colony Optimisation, see the course on SwarmIntelligence or the book by Dorigo and Stutzle [2004].


Evolutionary Algorithms

Key idea: Iteratively apply genetic operators mutation,recombination, selection to a population of candidate solutions.

Inspired by simple model of biological evolution:

I Mutation introduces random variation in the genetic materialof individuals.

I Recombination of genetic material during reproductionproduces o↵spring that combines features inherited from bothparents.

I Di↵erences in evolutionary fitness lead selection of genetictraits (‘survival of the fittest’).


Evolutionary Algorithm (EA):

determine initial population sp

While termination criterion is not satisfied:|| generate set spr of new candidate solutions|| by recombination|||||||| generate set spm of new candidate solutions|| from spr and sp by mutation|||||||| select new population sp fromb candidate solutions in sp, spr , and spm


Problem: Pure evolutionary algorithms often lackcapability of su�cient search intensification.

Solution: Apply subsidiary local search after initialisation,mutation and recombination.

) Memetic Algorithms (aka Genetic Local Search)


Memetic Algorithm (MA):

determine initial population sp

perform subsidiary local search on sp

While termination criterion is not satisfied:|| generate set spr of new candidate solutions|| by recombination|||| perform subsidiary local search on spr|||| generate set spm of new candidate solutions|| from spr and sp by mutation|||| perform subsidiary local search on spm|||| select new population sp fromb candidate solutions in sp, spr , and spm


Initialisation

I Often: independent, uninformed random picking fromgiven search space.

I But: can also use multiple runs of construction heuristic.

Recombination

I Typically repeatedly selects a set of parents from currentpopulation and generates o↵spring candidate solutions fromthese by means of recombination operator.

I Recombination operators are generally based on linearrepresentation of candidate solutions and piece togethero↵spring from fragments of parents.


Example: One-point binary crossover operator

Given two parent candidate solutions x1x2 . . . xn and y1y2 . . . yn:

1. choose index i from set {2, . . . , n} uniformly at random;

2. define o↵spring as x1 . . . xi�1yi . . . yn and y1 . . . yi�1xi . . . xn.

0 1 1 0 1 1 1 0 Parent 1

cut

Parent 2

Offspring 1

Offspring 2

1 0 0 0 1 0 1 0

0 1 1 0 1 0 1 0

1 0 0 0 1 1 1 0

Generalization: two-point, k-point, uniform crossover


Mutation

I Goal: Introduce relatively small perturbations in candidatesolutions in current population + o↵spring obtained fromrecombination.

I Typically, perturbations are applied stochastically andindependently to each candidate solution; amount ofperturbation is controlled by mutation rate.

I Can also use subsidiary selection function to determine subsetof candidate solutions to which mutation is applied.

I In the past, the role of mutation (as compared torecombination) in high-performance evolutionary algorithmshas been often underestimated [Back, 1996].


Selection

I Selection for variation: determines which of the individualcandidate solutions of the current population are chosen toundergo recombination and/or mutation

I Selection for survival: determines population for next cycle(generation) of the algorithm by selecting individual candidatesolutions from current population + new candidate solutionsobtained from recombination, mutation (+ subsidiary localsearch).

I Goal: Obtain population of high-quality solutions whilemaintaining population diversity.

I Selection is based on evaluation function (fitness) ofcandidate solutions such that better candidate solutions havea higher chance of ‘surviving’ the selection process.


Selection (general)

I Many selection schemes involve probabilistic choices, usingthe idea that better candidate solutions have a higherprobability of being chosen.

I examplesI roulette wheel selection (probability of selecting a candidate

solution s is proportional to its fitness value, g(s))

I tournament selection (choose best of k randomly sampledcandidate solutions)

I rank-based computation of selection probabilities

I the strength of the probabilistic bias determines the selectionpressure


Selection (survival)

I generational replacement versus overlapping populations;(extreme case, steady-state selection)

I It is often beneficial to use elitist selection strategies, whichensure that the best candidate solutions are always selected.

I probabilistic versus deterministic replacement

I quasi-deterministic replacement strategies implemented byclassical selections schemes from evolution strategies (aparticular type of EAs)

I �: number o↵spring, µ: number parent candidate solutions

I (µ,�) strategy: choose best µ of � > µ o↵spring

I (µ+ �) strategy: choose best µ of µ+ � candidate solutions


Subsidiary local search

I Often useful and necessary for obtaining high-qualitycandidate solutions.

I Typically consists of selecting some or all individuals inthe given population and applying an iterative improvementprocedure to each element of this set independently.


Example: A memetic algorithm for SAT (1)

I Search space: set of all truth assignments for propositionalvariables in given CNF formula F ; solution set: models of F ;use 1-flip neighbourhood relation; evaluation function:number of unsatisfied clauses in F .

I Note: truth assignments can be naturally represented as bitstrings.

I Use population of k truth assignments; initialise by(independent) Uninformed Random Picking.



I Recombination: Add o↵spring from n/2 (independent)one-point binary crossovers on pairs of randomly selectedassignments from population to current population(n = number of variables in F ).

I Mutation: Flip µ randomly chosen bits of each assignmentin current population (mutation rate µ: parameter ofthe algorithm); this corresponds to µ steps of UninformedRandom Walk; mutated individuals are added to currentpopulation.

I Selection: Selects the k best assignments from currentpopulation (simple elitist selection mechanism).



I Subsidiary local search: Applied after initialisation,recombination and mutation; performs iterative bestimprovement search on each individual assignmentindependently until local minimum is reached.

I Termination: upon finding model of F or after bound onnumber of cycles (generations) is reached.

Note: This algorithm does not reach state-of-the-art performance,but many variations are possible (few of which have been explored).


Problem representation and operators

I simplest choice of candidate solution representation: bitstrings

I advantage: application of simple recombination, mutationoperators

I problems with this arise in case of

I problem constraints (e.g. set covering, graph bi-partitioning)

I “richer” problem representations are much better suited (e.g.TSP)

I possible solutions

I application of representation- (and problem-) specificrecombination and mutation operators

I application of repair mechanisms to reestablish feasibility ofcandidate solutions


Memetic algorithm by Merz and Freisleben (MA-MF)

I one of the best studied MAs for the TSP

I first versions proposed in 1996 and further developed until2001

I main characteristics

I population initialisation by constructive searchI exploits an e↵ective LK implementationI specialised recombination operatorI restart operators in case the search is deemed to stagnateI standard selection and mutation operators


MA-MF: population initialisation

I each individual of initial population constructed by arandomised variant of the greedy heuristic

Step 1: choose n/4 edges by the following two stepsI select a vertex v 2 V uniformly at random

among those that are not yet in partial tourI insert shortest (second-shortest) feasible edge

incident to v with a probability of 2/3 (1/3)

Step 2: complete tour using the greedy heuristic

I locally optimise initial tours by LK


MA-MF: recombination

I various specialised crossover operators examined(distance-preserving crossover, greedy crossover)

I crossover operators generate feasible o↵spring

I best performance by greedy crossover that borrows ideas fromgreedy heuristic

I one o↵spring is generated from two parents:

1. copy fraction of pe

common edges to o↵spring2. add fraction of p

n

new short edges not contained in any of theparents

3. add fraction of pc

shortest edges from parents4. complete tour by greedy heuristic

I best performance for pe

= 1; µ/2 pairs of tours are chosenuniformly at random from population for recombination


MA-MF: mutation, selection, restart, results

I mutation by (usual) double-bridge move

I selection done by usual (µ+ �) strategy

I µ: population size, �: number of new o↵spring generatedI select µ lowest weight tours among µ+ � current tours for

next iterationI take care that no duplicate tours occur in population

I partial restart by strong mutation

I if average distance is below 10 or did not change for 30iterations, apply random k-exchange move (k = 0.1 · n) pluslocal search to all individuals except population-best one

I results: high solution quality reachable, though not fullycompetitive with state-of-the-art ILS algorithms for TSP


Memetic algorithm by Walters (MA-W)

I di↵ers in many aspects from other MAs for the TSP

I main di↵erences concern

I solution representation by nearest neighbour indexing insteadof permutation representation

I usage of general-purpose recombination operators that maygenerate infeasible o↵spring

I repair mechanism is used to restore valid tours from infeasibleo↵spring

I uses ”only” a 3–opt algorithm as subsidiary local search


MA-W: solution representation, initialisation, mutation

I solution representation through nearest neighbour indexing

I tour p represented as vector s := (s1, . . . , sn) such that si

= kif, and only if, the successor of vertex u

i

in p is kth nearestneighbour of u

i

I leads, however, to some redundancies for symmetric TSPs

I population initialisation by choosing randomly nearestneighbour indices

I three nearest neighbours selected with probability of0.45, 0.25, 0.15, respectively

I in remaining cases index between four and ten chosenuniformly at random

I mutation modifies nearest neighbour indices of randomlychosen vertices according to same probability distribution


MA-W: recombination, repair mechanism, results

I recombination is based on a slight variation of standardtwo-point crossover operator

I infeasible candidate solutions from crossover and mutation arerepaired

I repair mechanism tries to preserve as many edges as possibleand replaces an edge e by an edge e 0 such that |w(e)�w(e 0)|is minimal

I results are interesting considering that ”only” 3–opt was usedas subsidiary local search; however, worse than state-of-the-artILS algorithms


Tour merging

I can be seen as an extreme case of MAs

I exploits information collected by high-quality solutions fromvarious ILS runs in a two phases approach

I phase oneI generate a set T of very high quality tours for G = (V ,E ,w)I define subgraph G 0 = (V ,E 0,w 0), where E 0 contains all edges

in at least one t 2 T and w 0 is original w restricted to E 0

I phase twoI determine optimal tour in G 0

I Note: general-purpose or specialised algorithm that exploitcharacteristics of T are applicable

I very high quality solutions can be obtainedI optimal solution to TSPLIB instance d15112 in 22 days on a

500 MHz Alpha processorI new best-known solutions to instances brd14051 and d18512


Types of evolutionary algorithms (1)

I Genetic Algorithms (GAs) [Holland, 1975; Goldberg, 1989]:

I have been applied to a very broad range of (mostly discrete)combinatorial problems;

I often encode candidate solutions as bit strings of fixed length,which is now known to be disadvantagous for combinatorialproblems such as the TSP.

Note: There are some interesting theoretical results for GAs(e.g., Schema Theorem), but – as for SA – their practicalrelevance is rather limited.


Types of evolutionary algorithms (2)

I Evolution Strategies [Rechenberg, 1973; Schwefel, 1981]:

I orginally developed for (continuous) numerical optimisationproblems;

I operate on more natural representations of candidate solutions;

I use self-adaptation of perturbation strength achieved bymutation;

I typically use elitist deterministic selection.

I Evolutionary Programming [Fogel et al., 1966]:

I similar to Evolution Strategies (developed independently),but typically does not make use of recombination and usesstochastic selection based on tournament mechanisms.


variable neighbourhood search...

Documents