grasp – a speedy introduction · grasp – a speedy introduction thomas stidsen ... elements only...

1Thomas Stidsen

DTU-Management / Operations Research

GRASP – A speedy introductionThomas Stidsen

[email protected]

DTU-Management

Technical University of Denmark

2Thomas Stidsen


GRASPGRASP is an abbreviation for

Greedy

Randomized

Adaptive

Search

Procedure.It was “invented” by Feo and Resende in 1989.

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Overview of GRASPGRASP consists of 2 phases:

Greedy Randomized Adaptive phase(generating a solution).

Local Search (finding a local minimum).

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Greedy Randomized Adaptive phaseThe first phase consists of 3 parts:

1. Greedy algorithm.

2. Adaptive function.

3. Probabilistic selection.

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Greedy algorithmA greedy algorithm always makes the choicethat looks best at the moment. It makes alocally optimal choice in the hope that thischoice will lead to a globally optimal solution.

Greedy algorithms do not always yield optimalsolutions (eg. 0-1-knapsack), but in some casesit does (eg. Minimum spanning tree).

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Mathematical Model for Knapsack problemobjective: maximize

∑

i

ci · xi

s.t.∑

i

wi · xi ≤ W

xi ∈ 0, 1

The profit weight ratio:PF = ci

wi

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Greedy algorithm for 0-1 knapsackprocedure greedy_knapsack()

calculate profit vs. weight ratiowhile the smallest item can still fit in do

Add largest PF = ci

wi

elementUpdate remaining weight

return solution

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Example: 0-1-Knapsackproblem

item 1

item 2

item 3

knapsack

120

10 20 30

value: 60 100

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Example: 0-1-Knapsackproblem

knapsack knapsack

180

120

100

100

60 60

120

knapsack

220 160

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Greedy algorithm for the CLP ?How can we make a greedy algorithm for theChristmas lunch problem ? Lets remember acouple of facts:

We want to maximize the interest count.

A person at a table can only achieve a positiveinterest from other persons at a table.

If a table is empty and a person is put there,there is no gain ...

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Greedy algorithm for the CLPSo, we will make the greedy algorithm the followingway:

procedure greedy_clp()for each table doPlace one un-seated person randomly chosenwhile still room at the table do

Add the person with the largest interest gainreturn solution

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Probabilistic selectionIn the greedy algorithm the selection of the nextelement to add to the solution is (often !)deterministic.

The probabilistic selection process selectscandidates among which we choose the nextelement to be added to the presently partialsolution. The list of candidates is formallydenoted the Restricted Candidate List (RCL).

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Restricted Candidate ListConstruction:

Select the β best elements (β = 1: purelygreedy, β = n completely random).Selects the elements that are not more thanα% away from the best element. (α = 0:purely greedy, α = ∞: purely random).Select elements above an absolutethreshold of γ.

Selection:Equally probability.Weighted probability.

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Adaptive functionModify the function which guides the greedyalgorithm based on the element selected for thesolution we are constructing.

The construction is called dynamic in contrastto the static approach which assigns a score toelements only before starting the construction(example: TSP links).

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GRASP main codeprocedure grasp()

CurrentBest = 0while 〈 more time 〉

Solution = ConstructSolution()NewSolution = LocalSearch(Solution)if 〈 NewSolution better than CurrentBest 〉 then

CurrentBest = NewSolutionreturn CurrentBest

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Greedy Randomized solution constructionprocedure ConstructSolution()

Solution = ∅for 〈 solution construction not finished 〉

CandidateList = MakeCandidateList()s = SelectElement(CandidateList)Solution = Solution ∪sChangeGreedyFunction();

return Solution

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Additions to the GRASP schemeUse different adaptive functions during theexecution of the algorithm.

Settings of the Restricted Candidate Listdynamic.

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More informationThe website

http://www.graspheuristic.org

contains an annotated bibliography of GRASP.This is a good starting point for every form ofwork with the GRASP metaheuristic.

Object-oriented framework developed byAndreatta, Carvalho and Ribero.

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Pros and consPRO: Simple structure (means often easy toimplement),

PRO: If you can construct a greedy algorithm,extension to GRASP is often easy.

PRO: Can be used for HUGE optimizationproblems.

CONS: dependent on a “natural” greedyalgorithm, no escape from local optimum.

CONS: may easily re-discover the samesolution many times

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(Adaptive) Large Neighbourhood SearchA recently developed method is: LargeNeighbourhood Search (LNS) or Adaptive LargeNeighbourhood Search (ALNS).

LNS first suggested in 1998 by Shaw: "Usingconstraint programming and local searchmethods to solve vehicle routing problems".

ALNS first suggested in 2006 by Ropke andPisinger: "An adaptive large neighborhoodsearch heuristic

Both LNS and ALNS are similar to GRASP andhave shown VERY interesting results.

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LNS IThe basic idea in LNS is to use DESTROY andREPAIR methods to create new solutions (based onthe current solution):

A destroy method removes a part of thesolution ...

A repair method recreates the whole (feasible)solution.

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LNS IIThe basic idea in LNS is to use DESTROY andREPAIR methods to create new solutions (based onthe current solution):

Together destroy and repair methods form aneighbourhood, typically a very largeneighbourhood.

The destroy method is often stochastic, i.e. verysimply deleting a certain part of the solution.

The repair method is very often a kind of greedyconstructive algorithm.

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LNS III : Pseudo codeprocedure LNS()

Generate feasible solution x

xb = xrepeat

xt = r(d(x))if accept(xt,x) then x = xt

if c(xt) < c(xb)) then xb = xt

until time limitreturn xb

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ALNS IAdaptive Large Neighbourhood Search is basicallyan extension of LNS: Instead of having one destroyand one repair method, ALNS allows many differentdestroy and repair methods:

In each iteration a destroy and repair methodsis chosen probabilistically.

The chance for being chosen is adaptivelychosen so that successful repair and destroymethods are rewarded for their success.

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LNS II : Pseudo codeprocedure ALNS()

Generate feasible solution x

xb = x, ρ− = (1, ..., 1), ρ+ = (1, ..., 1)repeat

select destroy and repair methods d ∈ Ω−

and r ∈ Ω+ using ρ− and ρ+

xt = r(d(x))if accept(xt,x) then x = xt

if c(xt) < c(xb)) then xb = xt

update ρ− and ρ+

until time limitreturn xb

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How to select destroy and repairThe probability for choosing repair/destroy operatornumber j is:

φ− =ρ−

∑|Ω−|k=1

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How to adjust ρ− and ρ+ IFor every iteration exactly one destroy and onerepair method. The used methods (and only them)are given a value Ψ = MAX(ω1, ω2, ω3, ω4), whereω1 ≥ ω2 ≥ ω3 ≥ ω4.

ω1: reward if new solution is new global best

ω2: reward if new solution is better than thecurrent one

ω3: reward if new solution is accepted

ω4: "reward" if new solution is rejected

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How to adjust ρ− and ρ+ IIFinally, the new ρ− and ρ+ for the selected destroyand repair methods, ρ−

aand ρ+

a:

ρ−a

= λρ−a

+ (1 − λ)ρ−a

Where λ ∈ [0, 1]

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LNS and ALNS commentsBoth methods have shown excellent results.

Both methods can be considered to begeneralizations of GRASP (complete destroyand probabilistic repair ...)

A somewhat similar method "VariableNeighbourhood Search" is described in thebook ... but I much prefer the handed out articleby Roepke and Pisinger.

grasp – a speedy introduction · grasp – a speedy introduction thomas stidsen ... elements only...

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