metaheuristics in optimization1 panos m. pardalos university of florida ise dept., florida, usa...

Metaheuristics in Optimization 1

Panos M. Pardalos University of Florida

ISE Dept., Florida, USA

Workshop on the European Chapter on Metaheuristics and Large Scale Optimization

Vilnius, LithuaniaMay 19-21, 2005

Metaheuristics in Optimization

a


1. Quadratic Assignment & GRASP

2. Classical Metaheuristics

3. Parallelization of Metaheuristcs

4. Evaluation of Metaheuristics

5. Success Stories

6. Concluding Remarks

Outline

1. Quadratic Assignment & GRASP

2. Classical Metaheuristics

3. Parallelization of Metaheuristcs

4. Evaluation of Metaheuristics

5. Success Stories

6. Concluding Remarks

(joint work with Mauricio Resende and Claudio Meneses)


• Metaheuristics are high level procedures that coordinate simple heuristics, such as local search, to find solutions that are of better quality than those found by the simple heuristics alone.

• Examples: simulated annealing, genetic algorithms, tabu search, scatter search, variable neighborhood search, and GRASP.

Metaheuristics

• Metaheuristics are high level procedures that coordinate simple heuristics, such as local search, to find solutions that are of better quality than those found by the simple heuristics alone.

• Examples: simulated annealing, genetic algorithms, tabu search, scatter search, variable neighborhood search, and GRASP.


Quadratic assignment problem (QAP)

• Given N facilities f1,f2,…,fN and N locations l1,l2,…,lN

• Let AN×N = (ai,j) be a positive real matrix where ai,j is the flow between facilities fi and fj

• Let BN×N = (bi,j) be a positive real matrix where bi,j is the distance between locations li and lj



• Let p: {1,2,…,N} {1,2,…,N} be an assignment of the N facilities to the N locations

• Define the cost of assignment p to be

• QAP: Find a permutation vector p ∏N that minimizes the assignment cost:

N

1j p(j)p(i),ji,

N

1iba c(p)

min c(p): subject to p ∏N






N

1j p(j)p(i),ji,

N

1iba c(p)




l1 l2

l3

10

30 40

locations and distances

f1 f2

f3

1

510

facilities and flows

cost of assignment: 10×1+ 30×10 + 40×5 = 510



l1 l2

l3

10

30 40

f1 f2

f3

1

510


f1 f2

f3

1

510



swap locations of facilities f2 and f3



l1 l2

l3

10

30 40

f1 f3

f2

51

10

cost of assignment: 10×10+ 30×5 + 40×1 = 290Optimal!

f1 f2

f3

1

510


swap locations of facilities f1 and f3


GRASP for QAP

• GRASP multi-start metaheuristic: greedy randomized construction, followed by local search (Feo & Resende, 1989, 1995; Festa & Resende, 2002; Resende & Ribeiro, 2003)

• GRASP for QAP– Li, Pardalos, & Resende (1994): GRASP for QAP– Resende, Pardalos, & Li (1996): Fortran subroutines

for dense QAPs– Pardalos, Pitsoulis, & Resende (1997): Fortran

subroutines for sparse QAPs– Fleurent & Glover (1999): memory mechanism in

construction


repeat {

x = GreedyRandomizedConstruction();x = LocalSearch(x);

save x as x* if best so far;

}

return x*;

GRASP for QAP


Construction

• Stage 1: make two assignments {filk ; fjll}

• Stage 2: make remaining N–2 assignments of facilities to locations, one facility/location pair at a time


Stage 1 construction

• sort distances bi,j in increasing order: bi(1),j(1)≤bi(2),j(2) ≤ ≤ bi(N),j(N) .

• sort flows ak,l in decreasing order: ak(1),l(1)ak(2),l(2) ak(N),l(N) .

• sort products: ak(1),l(1) bi(1),j(1), ak(2),l(2) bi(2),j(2), …, ak(N),l(N)

bi(N),j(N)

• among smallest products, select ak(q),l(q) bi(q),j(q) at random: corresponding to assignments {fk(q)li(q) ; fl(q)lj(q)}






bi(N),j(N)




• If Ω = {(i1,k1),(i2,k2), …, (iq,kq)} are the q assignments made so far, then

• Cost of assigning fjll is

• Of all possible assignments, one is selected at random from the assignments having smallest costs and is added to Ω

ki,

lk,ji,lj, bac






ki,

lk,ji,lj, bac






ki,

lk,ji,lj, bac

Sped up in Pardalos, Pitsoulis, & Resende (1997) forQAPs with sparse A or B matrices.


Swap based local search

a) For all pairs of assignments {filk ; fjll}, test if swapped assignment {fill ; fjlk} improves solution.

b) If so, make swap and return to step (a)


Swap based local search

a) For all pairs of assignments {filk ; fjll}, test if swapped assignment {fill ; fjlk} improves solution.

b) If so, make swap and return to step (a)

repeat (a)-(b) until no swap improves current solution


Path-relinking• Path-relinking:

– Intensification strategy exploring trajectories connecting elite solutions: Glover (1996)

– Originally proposed in the context of tabu search and scatter search.

– Paths in the solution space leading to other elite solutions are explored in the search for better solutions:

• selection of moves that introduce attributes of the guiding solution into the current solution


Path-relinking

• Exploration of trajectories that connect high quality (elite) solutions:

initialsolution

guidingsolution

path in the neighborhood of solutions


Path-relinking• Path is generated by selecting moves

that introduce in the initial solution attributes of the guiding solution.

• At each step, all moves that incorporate attributes of the guiding solution are evaluated and the best move is selected:

initialsolution

guiding solution





initialsolution

guiding solution


Combine solutions x and y

(x,y): symmetric difference between x and y

while ( |(x,y)| > 0 ) {

-evaluate moves corresponding in (x,y)

-make best move

-update (x,y)

}

Path-relinking

x

y


GRASP with path-relinking• Originally used by Laguna and Martí (1999).• Maintains a set of elite solutions found

during GRASP iterations.• After each GRASP iteration (construction

and local search):– Use GRASP solution as initial solution. – Select an elite solution uniformly at random:

guiding solution.– Perform path-relinking between these two

solutions.






solutions.


GRASP with path-relinkingRepeat for Max_Iterations:

Construct a greedy randomized solution.

Use local search to improve the constructed solution.

Apply path-relinking to further improve the solution.

Update the pool of elite solutions.

Update the best solution found.


P-R for QAP (permutation vectors)


If swap improves solution: local search is applied

Path-relinking for QAP

initialsolution

guidingsolution

local minlocal minIf local min

improvesincumbent, it is saved.


Results of path relinking: S*


initialsolution

guidingsolution


S*

If c(S*) < min {c(S), c(T)}, and c(S*) ≤ c(Si), for i=1,…,N,i.e. S* is best solution in path, then S* is returned.

ST

S0

S1

S2

S3SN


initialsolution

guidingsolution


S*

S T

S0

Si–1

Si

Si+1

SN

Si is a local minimum w.r.t. PR: c(Si) < c(Si–1) and c(Si) < c(Si+1), for all i=1,…,N.

If path-relinking does not improve (S,T), then if Si is a best local min w.r.t. PR: return S* = Si

If no local min exists, return S*=argmin{S,T}


PR pool management

• S* is candidate for inclusion in pool of elite solutions (P)

• If c(S*) < c(Se), for all Se P, then S* is put in P• Else, if c(S*) < max{c(Se), Se P} and |

(S*,Se)| 3, for all Se P, then S* is put in P• If pool is full, remove

argmin {|(S*,Se)|, Se P s.t. c(Se) c(S*)}


PR pool management






PR pool management

S is initial solution for path-relinking: favor choice of target solution T with large symmetric difference with S.

This leads to longer paths in path-relinking.

PR

ee

|R)(S,||)S(S,|

)p(S

Probability of choosing Se P:


Experimental results

• Compare GRASP with and without path-relinking.

• New GRASP code in C outperforms old Fortran codes: we use same code to compare algorithms

• All QAPLIB (Burkhard, Karisch, & Rendl, 1991) instances of size N ≤ 40

• 100 independent runs of each algorithm, recording CPU time to find the best known solution for instance


• SGI Challenge computer (196 MHz R10000 processors (28) and 7 Gb memory)

• Single processor used for each run• GRASP RCL parameter chosen at random in

interval [0,1] at each GRASP iteration.• Size of elite set: 30• Path-relinking done in both directions (S to T to

S)• Care taken to ensure that GRASP and GRASP

with path-relinking iterations are in sync



0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

pro

babili

ty

time to target solution value (seconds)

Random variable time-to-target-solution value fits a two-parameter exponential distribution (Aiex, Resende, & Ribeiro, 2002).

Time-to-target-value plots

Sort times such that t1 ≤ t2 ≤ ∙∙∙ ≤ t100 and plot{ti,pi}, for i=1,…,N, wherepi = (i–.5)/100



0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

pro

babili

ty

time to target solution value (seconds)

In 80% of trials targetsolution is found in less than 1.4 s

Probability of finding target solution in less than 1 s is about 70%.



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 100 200 300 400 500 600 700

cu

mu

lative

pro

ba

bili

ty

time to target solution

ALG 1 ALG 2

For a given time, compare probabilities of finding targetsolution in at most that time.

For a given probability, compare times required to find with given probability.

We say ALG 1 is faster than ALG 2


C.E. Nugent, T.E. Vollmann and J. Ruml [1968]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2000 4000 6000 8000 10000 12000 14000

cu

mu

lative

pro

ba

bility

time to target value (seconds on an SGI Challenge 196MHz R10000)

prob: nug30

GRASP with PRGRASP0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100 150 200 250 300

cu

mu

lative

pro

ba

bility


prob: nug25

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30

cu

mu

lative

pro

ba

bility


prob: nug20

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2

cu

mu

lative

pro

ba

bility


prob: nug12

GRASP with PRGRASP

nug12 nug20

nug25 nug30


E.D. Taillard [1991, 1994]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25

cu

mu

lative

pro

ba

bility


prob: tai15a

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30 35 40 45

cu

mu

lative

pro

ba

bility


prob: tai17a

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 200 400 600 800 1000 1200

cu

mu

lative

pro

ba

bility


prob: tai20a

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5000 10000 15000 20000 25000 30000 35000 40000 45000

cu

mu

lative

pro

ba

bility


prob: tai25a

GRASP with PRGRASP

tai15a tai17a

tai20a tai25a


Y. Li and P.M. Pardalos [1992]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9

cu

mu

lative

pro

ba

bility


prob: lipa20a

GRASP with PRGRASP 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 100 200 300 400 500 600 700

cu

mu

lative

pro

ba

bility


prob: lipa30a

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20000 40000 60000 80000 100000 120000 140000 160000

cu

mu

lative

pro

ba

bility


prob: lipa40a

GRASP with PRGRASP

lipa20alipa30a

lipa40a


U.W. Thonemann and A. Bölte [1994]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 100000 200000 300000 400000 500000 600000

cu

mu

lative

pro

ba

bility


prob: tho40

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 500 1000 1500 2000 2500 3000 3500

cu

mu

lative

pro

ba

bility


prob: tho30

GRASP with PRGRASP

tho30

tho40


L. Steinberg [1961]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20000 40000 60000 80000 100000 120000 140000 160000 180000

cu

mu

lative

pro

ba

bility


prob: ste36c

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50000 100000 150000 200000 250000 300000 350000 400000

cu

mu

lative

pro

ba

bility


prob: ste36a

GRASP with PRGRASP

ste36a

ste36c

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1000 2000 3000 4000 5000 6000 7000 8000

cu

mu

lative

pro

ba

bili

ty


prob: ste36b

GRASP with PRGRASP

ste36b


M. Scriabin and R.C. Vergin [1975]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

cu

mu

lative

pro

ba

bility


prob: scr12

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

cu

mu

lative

pro

ba

bility


prob: scr15

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60 70 80 90 100

cu

mu

lative

pro

ba

bility


prob: scr20

GRASP with PRGRASP

scr12 scr15

scr20


S.W. Hadley, F. Rendl and H. Wolkowicz

[1992]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

cu

mu

lative

pro

ba

bility


prob: had14

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3

cu

mu

lative

pro

ba

bility


prob: had16

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

cu

mu

lative

pro

ba

bility


prob: had18

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5

cu

mu

lative

pro

ba

bility


prob: had20

GRASP with PRGRASP

had14 had16

had18 had20


R.E. Burkard and J. Offermann [1977]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30 35

cu

mu

lative

pro

ba

bili

ty


prob: bur26a

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100 120 140

cu

mu

lative

pro

ba

bili

ty


prob: bur26b

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60 70 80

cu

mu

lative

pro

ba

bili

ty


prob: bur26c

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60 70 80 90

cu

mu

lative

pro

ba

bili

ty


prob: bur26d

GRASP with PRGRASP

bur26a bur26b

bur26c bur26d


N. Christofides and E. Benavent [1989]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100 150 200 250

cu

mu

lative

pro

ba

bili

ty


prob: chr18a

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

cu

mu

lative

pro

ba

bili

ty


prob: chr20a

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2000 4000 6000 8000 10000 12000 14000 16000

cu

mu

lative

pro

ba

bili

ty


prob: chr22a

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2000 4000 6000 8000 10000 12000 14000 16000

cu

mu

lative

pro

ba

bili

ty


prob: chr25a

GRASP with PRGRASP

chr18a chr20a

chr22a chr25a


C. Roucairol [1987]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

cu

mu

lative

pro

ba

bili

ty


prob: rou12

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5 4

cu

mu

lative

pro

ba

bili

ty


prob: rou15

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 100 200 300 400 500 600 700 800

cu

mu

lative

pro

ba

bili

ty


prob: rou20

GRASP with PRGRASP

rou12 rou15

rou20


J. Krarup and P.M. Pruzan [1978]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 100 200 300 400 500 600 700 800 900

cu

mu

lative

pro

ba

bili

ty


prob: kra30a

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1000 2000 3000 4000 5000 6000 7000

cu

mu

lative

pro

ba

bili

ty


prob: kra30b

GRASP with PRGRASP

kra30a

kra30b


B. Eschermann and H.J. Wunderlich [1990]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1000 2000 3000 4000 5000 6000

cu

mu

lative

pro

ba

bili

ty


prob: esc32a

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30

cu

mu

lative

pro

ba

bili

ty


prob: esc32b

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5

cu

mu

lative

pro

ba

bili

ty


prob: esc32d

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6

cu

mu

lative

pro

ba

bili

ty


prob: esc32h

GRASP with PRGRASP

esc32a esc32b

esc32d esc32h


B. Eschermann and H.J. Wunderlich [1990]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.02 0.04 0.06 0.08 0.1 0.12

cu

mu

lative

pro

ba

bili

ty


prob: esc32c

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06

cu

mu

lative

pro

ba

bili

ty


prob: esc32e

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

cu

mu

lative

pro

ba

bili

ty


prob: esc32f

GRASP with PRGRASP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06

cu

mu

lative

pro

ba

bili

ty


prob: esc32g

GRASP with PRGRASP

esc32c esc32e

esc32f esc32g


Remarks

• New heuristic for the QAP is described.• Path-relinking shown to improve performance of

GRASP on almost all instances.• Experimental results and code are available at

http://www.research.att.com/~mgcr/exp/gqapspr


Remarks





Classical Metaheuristics

• Simulated Annealing• Genetic Algorithms• Memetic Algorithms• Tabu Search• GRASP• Variable Neighborhood Search• etc

(see Handbook of Applied Optimization, P. M. Pardalos and M. G. Resende, Oxford University Press, Inc., 2002)

http://www.oup.com/


Input: A problem instanceOutput: A (sub-optimal) solution

1. Generate an initial solution at random and initialize the temperature T2. While (T > 0) do (a) While (thermal equilibrium not reached) do (i) Generate a neighbor state at random and evaluate the change in energy level ΔE (ii) If ΔE < 0, update current state with new state (iii) If ΔE < 0, update current state with new state with probability (b) Decrease temperature T according to annealing schedule

3. Output the solution having the lowest energy

Simulated AnnealingInput: A problem instanceOutput: A (sub-optimal) solution

1. Generate an initial solution at random and initialize the temperature T2. While (T > 0) do (a) While (thermal equilibrium not reached) do (i) Generate a neighbor state at random and evaluate the change in energy level ΔE (ii) If ΔE < 0, update current state with new state (iii) If ΔE < 0, update current state with new state with probability (b) Decrease temperature T according to annealing schedule

3. Output the solution having the lowest energy

TK

E

Be



1. t=0, Initialize P(t), evaluate the fitness of the individuals in P(t)2. While (termination condition is not satisfied) do (i) t = t+1 (ii) Select P(t), recombine P(t) and evaluate P(t)

3. Output the best solution among all the population as the (sub-optimal) solution


1. t=0, Initialize P(t), evaluate the fitness of the individuals in P(t)2. While (termination condition is not satisfied) do (i) t = t+1 (ii) Select P(t), recombine P(t) and evaluate P(t)


Genetic Algorithms



1. t=0, Initialize P(t), evaluate the fitness of the individuals in P(t)2. While (termination condition is not satisfied) do (i) t = t+1 (ii) Select P(t), recombine P(t), perform local search on each individual of P(t), evaluate P(t)



1. t=0, Initialize P(t), evaluate the fitness of the individuals in P(t)2. While (termination condition is not satisfied) do (i) t = t+1 (ii) Select P(t), recombine P(t), perform local search on each individual of P(t), evaluate P(t)


Memetic Algorithms


Input: A problem instanceOutput: A (sub-optimal) solution1. Initialization

(i) Generate an initial solution x and set x*=x(ii) Initialize the tabu list T=Ø(ii) Set iteration cunters k=0 and m=0

2. While (N(x)\T ≠ Ø) do (i) k=k+1; m=m+1 (ii) Select x as the best solution from set N(x)\T

(iii) If f(x) < f(x*) then update x*=x and set m=0

(iv) if k=kmax or m=mmax go to step 3

3. Output the best solution found x*

Input: A problem instanceOutput: A (sub-optimal) solution1. Initialization

(i) Generate an initial solution x and set x*=x(ii) Initialize the tabu list T=Ø(ii) Set iteration cunters k=0 and m=0

2. While (N(x)\T ≠ Ø) do (i) k=k+1; m=m+1 (ii) Select x as the best solution from set N(x)\T

(iii) If f(x) < f(x*) then update x*=x and set m=0

(iv) if k=kmax or m=mmax go to step 3

3. Output the best solution found x*

Tabu Search


Input: A problem instance

Output: A (sub-optimal) solution

1. Repeat for Max_Iterations

(i) Construct a greedy randomized solution

(ii) Use local search to improve the constructed

solution

(ii) Update the best solution found


GRASPInput: A problem instance

Output: A (sub-optimal) solution

1. Repeat for Max_Iterations

(i) Construct a greedy randomized solution

(ii) Use local search to improve the constructed

solution

(ii) Update the best solution found




1. Initialization:

(i) Select the set of neighborhood structures Nk, k=1,…,kmax, that will be used in the search(ii) Find an initial solution x(iii) Choose a stopping condition

2. Repeat until stopping condition is met (i) k=1

(ii) While (k ≤ kmax) do

(a) Shaking: Generate a point y at random from Nk(x) (b) Local Search: Apply some local search method with y as initial solution; Let z be the local optimum (c) Move or not: If z is better than the incumbent, move there (x = z), and set k=1; otherwise set k=k+1

3. Output the incumbent solution


1. Initialization:

(i) Select the set of neighborhood structures Nk, k=1,…,kmax, that will be used in the search(ii) Find an initial solution x(iii) Choose a stopping condition

2. Repeat until stopping condition is met (i) k=1

(ii) While (k ≤ kmax) do

(a) Shaking: Generate a point y at random from Nk(x) (b) Local Search: Apply some local search method with y as initial solution; Let z be the local optimum (c) Move or not: If z is better than the incumbent, move there (x = z), and set k=1; otherwise set k=k+1

3. Output the incumbent solution

VNS (Variable Neighborhood Search)


• Construction phase: greediness + randomization– Builds a feasible solution:

• Use greediness to build restricted candidate list and apply randomness to select an element from the list.

• Use randomness to build restricted candidate list and apply greediness to select an element from the list.

• Local search: search in the current neighborhood until a local optimum is found– Solutions generated by the construction procedure are not

necessarily optimal:• Effectiveness of local search depends on: neighborhood

structure, search strategy, and fast evaluation of neighbors, but also on the construction procedure itself.

GRASP in more detaitls


• Greedy Randomized Construction:– Solution – Evaluate incremental costs of candidate

elements– While Solution is not complete do:

• Build restricted candidate list (RCL)• Select an element s from RCL at random• Solution Solution {s}• Reevaluate the incremental costs.

endwhile

Construction phase


Construction phase• Minimization problem

• Basic construction procedure: – Greedy function c(e): incremental cost

associated with the incorporation of element e into the current partial solution under construction

– cmin (resp. cmax): smallest (resp. largest) incremental cost

– RCL made up by the elements with the smallest incremental costs.


Construction phase• Cardinality-based construction:

– p elements with the smallest incremental costs

• Quality-based construction: – Parameter defines the quality of the elements

in RCL.– RCL contains elements with incremental cost

cmin c(e) cmin + (cmax –cmin) = 0 : pure greedy construction = 1 : pure randomized construction

• Select at random from RCL using uniform probability distribution


5

10

15

20

0 0.2 0.4 0.6 0.8 1

tim

e (

seco

nd

s)

for

10

00

ite

ratio

ns

RCL parameter alpha

total CPU time

local search CPU time

Illustrative results: RCL parameter

weighted MAX-SAT instance

random greedyRCL parameter α

SGI Challenge 196 MHz


0

5

10

15

20

25

30

35

40

45

0 0.2 0.4 0.6 0.8 1 400000

405000

410000

415000

420000

425000

430000

435000

440000

445000

450000

best solution

average solution

time

tim

e (

seco

nd

s) f

or

10

00

ite

rati

on

s

solu

tion

valu

e

RCL parameter α

Illustrative results: RCL parameter

random greedy

weighted MAX-SAT instance: 100 variables and 850 clauses



Path-relinking

• Exploration of trajectories that connect high quality (elite) solutions:

initialsolution

guidingsolution






initialsolution

guiding solution


Elite solutions x and y

(x,y): symmetric difference between x and y

while ( |(x,y)| > 0 ) {

evaluate moves corresponding in (x,y) make best move

update (x,y)

}

Path-relinking


GRASP: 3-index assignment (AP3)

cost = 10

Complete tripartite graph:Each triangle made up ofthree distinctly colored nodes has a cost.

cost = 5

AP3: Find a set of trianglessuch that each node appearsin exactly one triangle and thesum of the costs of the triangles is minimized.


3-index assignment (AP3)

• Construction: Solution is built by selecting n triplets, one at a time, biased by triplet costs.

• Local search: Explores O(n2) size neighborhood of current solution, moving to better solution if one is foundAiex, Pardalos, Resende, & Toraldo (2003)



• Path relinking is done between:– Initial solution

S = { (1, j1S, k1S ), (2, j2S, k2

S ), …, (n, jnS, knS

) }– Guiding solution

T = { (1, j1T, k1T ), (2, j2T, k2

T ), …, (n, jnT, kn

T ) }


GRASP with path-relinking• Originally used by Laguna and Martí (1999).• Maintains a set of elite solutions found during GRASP

iterations.• After each GRASP iteration (construction and local

search):– Use GRASP solution as initial solution. – Select an elite solution uniformly at random: guiding

solution (may also be selected with probabilities proportional to the symmetric difference w.r.t. the initial solution).

– Perform path-relinking between these two solutions.


GRASP with path-relinking• Repeat for Max_Iterations:

– Construct a greedy randomized solution.– Use local search to improve the constructed

solution.– Apply path-relinking to further improve the

solution.– Update the pool of elite solutions.– Update the best solution found.


GRASP with path-relinking• Variants: trade-offs between computation time and

solution quality– Explore different trajectories (e.g. backward, forward):

better start from the best, neighborhood of the initial solution is fully explored!

– Explore both trajectories: twice as much the time, often with marginal improvements only!

– Do not apply PR at every iteration, but instead only periodically: similar to filtering during local search.

– Truncate the search, do not follow the full trajectory.– May also be applied as a post-optimization step to all

pairs of elite solutions.


GRASP with path-relinking• Successful applications:

1) Prize-collecting minimum Steiner tree problem: Canuto, Resende, & Ribeiro (2001) (e.g. improved all solutions found by approximation algorithm of Goemans & Williamson)

2) Minimum Steiner tree problem: Ribeiro, Uchoa, & Werneck (2002) (e.g., best known results for open problems in series dv640 of the SteinLib)

3) p-median: Resende & Werneck (2002) (e.g., best known solutions for problems in literature)


GRASP with path-relinking• Successful applications (cont’d):

4) Capacitated minimum spanning tree:Souza, Duhamel, & Ribeiro (2002) (e.g., best known results for largest problems with 160 nodes)

5) 2-path network design: Ribeiro & Rosseti (2002) (better solutions than greedy heuristic)

6) Max-Cut: Festa, Pardalos, Resende, & Ribeiro (2002) (e.g., best known results for several instances)

7) Quadratic assignment: Oliveira, Pardalos, & Resende (2003)


GRASP with path-relinking• Successful applications (cont’d):

8) Job-shop scheduling: Aiex, Binato, & Resende (2003)

9) Three-index assignment problem: Aiex, Resende, Pardalos, & Toraldo (2003)

10) PVC routing: Resende & Ribeiro (2003)

11) Phylogenetic trees: Ribeiro & Vianna (2003)


GRASP with path-relinking

• P is a set (pool) of elite solutions.

• Each iteration of first |P| GRASP iterations adds one solution to P (if different from others).

• After that: solution x is promoted to P if:– x is better than best solution in P.– x is not better than best solution in P, but is

better than worst and is sufficiently different from all solutions in P.


• GRASP is easy to implement in parallel:

– parallelization by problem decomposition• Feo, R., & Smith (1994)

– iteration parallelization• Pardalos, Pitsoulis, & R. (1995); Pardalos, Pitsoulis, &

R. (1996)• Alvim (1998); Martins & Ribeiro (1998)• Murphey, Pardalos, & Pitsoulis (1998)• R. (1998); Martins, R., & Ribeiro (1999)• Aiex, Pardalos, R., & Toraldo (2000)

• GRASP is easy to implement in parallel:

– parallelization by problem decomposition• Feo, R., & Smith (1994)

– iteration parallelization• Pardalos, Pitsoulis, & R. (1995); Pardalos, Pitsoulis, &

R. (1996)• Alvim (1998); Martins & Ribeiro (1998)• Murphey, Pardalos, & Pitsoulis (1998)• R. (1998); Martins, R., & Ribeiro (1999)• Aiex, Pardalos, R., & Toraldo (2000)

Parallelization of GRASP


Parallel independent implementation

• Parallelism in metaheuristics: robustnessCung, Martins, Ribeiro, & Roucairo (2001)

• Multiple-walk independent-thread strategy: – p processors available– Iterations evenly distributed over p processors– Each processor keeps a copy of data and algorithms. – One processor acts as the master handling seeds, data, and

iteration counter, besides performing GRASP iterations.– Each processor performs Max_Iterations/p iterations.


Parallel independent implementation

seed(1) seed(2) seed(3) seed(4) seed(p-1)

Best solution is sent to the master.

1 2 3 4 p-1Elite Elite Elite Elite Elite

Elite

pseed(p)


Parallel cooperative implementation

• Multiple-walk cooperative-thread strategy: – p processors available– Iterations evenly distributed over p-1 processors– Each processor has a copy of data and algorithms.– One processor acts as the master handling seeds,

data, and iteration counter and handles the pool of elite solutions, but does not perform GRASP iterations.

– Each processor performs Max_Iterations/(p–1) iterations.


Parallel cooperative implementation

2

Elite

1

p3

Elite solutions are stored in a centralized pool.Master

Slave SlaveSlave


Cooperative vs. independent strategies (for 3AP)

• Same instance: 15 runs with different seeds, 3200 iterations

• Pool is poorer when fewer GRASP iterations are done and solution quality deteriorates

procs best avg. best avg.

1 673 678.6 - -

2 676 680.4 676 681.6

4 680 685.1 673 681.2

8 687 690.3 676 683.1

16 692 699.1 674 682.3

32 702 708.5 678 684.8

Independent Cooperative


5

10

15

20

25

1 2 4 8 16

ave

rage s

peed-u

p

number of processors

independentcooperative

linear speedup

Speedup on 3-index assignment: bs24




Evaluation of Heuristics

• Experimental design

- problem instances

- problem characteristics of interest (e.g.,

instance size, density, etc.)

- upper/lower/optimal values


Evaluation of Heuristics (cont.)

• Sources of test instances

- Real data sets

It is easy to obtain real data sets

- Random variants of real data sets

The structure of the instance is preserved

(e.g., graph), but details are changed (e.g.,

distances, costs)



- Test Problem Libraries - Test problem collections with “best known” solution

- Test problem generators with known optimal

solutions (e.g., QAP generators, Maximum Clique,

Steiner Tree Problems, etc)


Evaluation of Heuristics (cont.)Test problem generators with known optimal solutions (cont.)

• C.A. Floudas, P.M. Pardalos, C.S. Adjiman, W.R. Esposito, Z. Gumus, S.T. Harding, J.L. Klepeis, C.A. Meyer, and C.A. Schweiger, Handbook of Test Problems for Local and Global Optimization, Kluwer Academic Publishers, (1999).

• C.A. Floudas and P.M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms, Springer-Verlag, Lecture Notes in Computer Science 455 (1990).

• J. Hasselberg, P.M. Pardalos and G. Vairaktarakis, Test case generators and computational results for the maximum clique problem, Journal of Global Optimization 3 (1993), pp. 463-482.

• B. Khoury, P.M. Pardalos and D.-Z. Du, A test problem generator for the steiner problem in graphs, ACM Transactions on Mathematical Software, Vol. 19, No. 4 (1993), pp. 509-522.

• Y. Li and P.M. Pardalos, Generating quadratic assignment test problems with known optimal permutations, Computational Optimization and Applications Vol. 1, No. 2 (1992), pp. 163-184.

• P. Pardalos, "Generation of Large-Scale Quadratic Programs", ACM Transactions on Mathematical Software, Vol. 13, No. 2, p. 133.

• P.M. Pardalos, Construction of test problems in quadratic bivalent programming, ACM Transactions on Mathematical Software, Vol. 17, No. 1 (1991), pp. 74-87.

• P.M. Pardalos, Generation of large-scale quadratic programs for use as global optimization test problems, ACM Transactions on Mathematical Software, Vol. 13, No. 2 (1987), pp. 133-137.



- Random generated instances (quickest and easiest way to obtain supply of test

instances)



• Performance measurement

- Time (most used, but difficult to assess

due to differences among computers)

- Solution Quality



• Solution Quality

- Exact solutions of small instances

For “small” instances verify results with

exact algorithms

- Lower and upper bounds

In many cases the problem of finding good bounds

is as difficult as solving the original problem



• Space covering techniques


Success Stories

The success of metaheuristics can be seen by the numerous applications for which they have been applied.

Examples:• Scheduling, routing, logic, partitioning• location• graph theoretic• QAP & other assignment problems• miscellaneous problems


Concluding Remarks

• Metaheuristics have been shown to perform well in practice

• Many times the globally optimal solution is found but there is no “certificate of optimality”

• Large problem instances can be solved implementing metaheuristics in parallel

• It seems it is the most practical way to deal with massive data set


References

• Handbook of Applied Optimization Edited by Panos M. Pardalos and Mauricio G. C. Resende,

Oxford University Press, Inc., 2002

• Handbook of Massive Data SetsSeries: Massive Computing, Vol. 4 Edited by J. Abello, P.M. Pardalos, M.G. Resende,

Kluwer Academic Publishers, 2002.

http://www.ise.ufl.edu/pardalos



http://www.research.att.com/%7Emgcr

http://www.research.att.com/%7Emgcr

http://www.oup.com/

http://www.springeronline.com/sgw/cda/frontpage/0,11855,5-146-69-33113779-0,00.html


THANK YOU ALL.

metaheuristics in optimization1 panos m. pardalos university of florida ise dept., florida, usa...

Documents