Lecture29

Sandeep Kumar Poonia
Algorithms
Local Search Methods
By: Sandeep Kumar Poonia
Asst. Professor, Jagannath University, Jaipur
6/15/2013

Genetic algorithms
• Genetic algorithms is based on the concepts
from population genetics and evolution theory.
The algorithm are constructed to optimize
fitness of a population of elements through
crossover (recombination) and mutation
(perturbation) operations on their genes.
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The Ant algorithm is based on the observation of real ant’s
behavior. Ants can coordinate their activities via stigmergy, a
way of indirect communication through the modification of the
environment. The main idea of ant algorithm is to use self-
organizing principles of artificial agents which collaborate to
solve the problems.
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The Ant algorithm

Tabu search
• Tabu search is a deterministic iterative
improvement local search method with a
possibility to accept worse-cost local solution
in order to escape from local optimum. The set
of legal local solutions are restricted by a tabu
list which is designed to present from going
back to the recently visited solutions. The set
of solutions in tabe list are not accepted in the
next iteration.
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Particle Swarm Optimization (PSO) algorithm is a population-
based stochastic optimization method proposed by James
Kennedy and R. C. Eberhart in 1995. It is motivated by social
behavior of organisms such as bird flocking and fish schooling.
In the PSO algorithm, the potential solutions called particles, are
flown in the problem hyperspace. Change of position of a
particle is called velocity. The particle changes their position
with time. During flight, particle’s velocity is stochastically
accelerated toward its previous best position and toward a
neighborhood best solution. PSO has been successfully applied
to solve various optimization problems, artificial neural network
training, fuzzy system control, and others.
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1
The Bees Algorithm is a new population-based search algorithm,
first developed in 2005 by Pham DT etc. [1] and Karaboga.D [2]
independently. The algorithm mimics the food foraging
behaviour of swarms of honey bees. In its basic version, the
algorithm performs a kind of neighbourhood search combined
with random search and can be used for optimization problems.
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Contents
1 Introduction
2 Genetic Algorithms
3 Non-linear Programming Problems
4 Foundation of Genetic Algorithms
5 Metaheuristic Algorithms
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6. Simulated Annealing
7. Tabu Search
8. Ant Algorithms
9. Particle Swarm Optimization
10. Bee Algorithms
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11. The Other Bio-Inspired Metaheuristic Algorithms
12. Traveling Salesman Problems
13. Knapsack Problems
14. Set Covering Problems
15. Minimum Spanning Tree Problems
16. Flow Shop Scheduling Problems
17. Job Shop and Open Shop Scheduling Problems
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18. Bin Packing Problems
19. Data Mining
20. Quadratic Assignment Problems
21. Reliability Optimization Problems
22. Layout and Location Problems
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1. The Computational complexity
We do not expect NP-hard (and NP-complete) problems to be
solved in polynomial steps.
Ambati et al. (1991): An evolution algorithm that can achieve
heuristic solutions 25% worse than the expected optimal solution on
random Traveling Salesman Problem in O(NlogN) time.
Fogel (1993): An evolution algorithm that can achieve heuristic
solutions 10% worse than the expected optimal solution on random
Traveling Salesman Problem in O(N2) time.
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goal
state
current
state
initial
state
Algorithm – Problem Solving
2. Search Spaces (Solution Space)
How to do efficient search in the state space?
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1
1. To obtain a good initial state.
2. Goal identification (identify the optimal condition).
3. Control the search process.
For some problems, we are not able to identify the goal state.
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9
Evolution Process
01
02
0k
S
S
S

11
12
1k
S
S
S

21
22
2k
S
S
S

1
2
n
n
nk
S
S
S

. . . .  
Initial
Population
s
1st
Generation
2nd
Generation
n-th
Generation
. . . .
Genetic Algorithm, Ant Algorithm, Particle Swarm Optimization,
Bee Algorithm, Firefly Algorithm, . . .
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Example 1: Simplex algorithm for linear program
Goal identification: primal feasible & dual feasible condition
Goal identification:
1. The goal state can be identified. (e.g. linear program, convex program )
2. The goal state cannot be identified. (e.g. traveling salesman problem,
quadratic assignment problem, knapsack problem, scheduling
problems, . . . )
Control the search process: moving to improved adjacent basic
solution
Example 2: Steepest descent algorithm for convex program
Goal identification: Karash-Kuhn-Tucker condition
Control the search process: moving to the steepest descent
direction ( minimization problem)
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Example 4: Ant algorithm for the Traveling Salesman Problem
Goal identification: The optimal condition can not be identified
efficiently. Using heuristic rules to identify approximate solution.
Control the search process: pheromone & state transition
probability
Example 3: Genetic algorithm for the Traveling Salesman Problem
Goal identification: The optimal condition can not be identified
efficiently. Using heuristic rules to identify approximate solution.
Control the search process: Crossover, mutation and selection
operations.
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A state space of a Traveling Salesman Problem with n = 4.
(A X X X)
(A B X X)
(A C X X)
(A D X X)
(A B C D)
(A B D C)
(A C B D)
(A C D B)
(A D B C)
(A D C B)
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1









searchincomplete
space)theinsearch(partialSearchLocal
searchcomplete
sapce)theinsearchc(systematiSearchExact
MethodsSearch
3. Search Methods
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Exact search methods:
Advantage: To produce an optimal solution and It is able to
detect that a given problem has no feasible solution.
Disadvantage: It is time-consuming. For example, it is infeasible
for real-time problems.
Local search methods:
Advantage: It is time-efficient and easy to write a program.
Disadvantage: It may not produce an optimal solution and is not
able to detect that a given problem has no feasible
solution.
Traversal Search, Backtracking Algorithm, Branch and Bound
Algorithm, Dynamic Programming Method, etc.
Traditional Local search does not provide a mechanism for the
search to escape from a local optimum. The goal of local search is
find a solution which is as close as possible to the optimum.Sandeep Kumar Poonia
6/15/2013

local search
• local search is a metaheuristic method for solving
computationally hard optimization problems.
• Local search can be used on problems that can be
formulated as finding a solution maximizing a
criterion among a number of candidate solutions.
• Local search algorithms move from solution to
solution in the space of candidate solutions (the
search space) by applying local changes, until a
solution deemed optimal is found or a time bound
is elapsed.
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local search
• A local search algorithm starts from a
candidate solution and then iteratively
moves to a neighbor solution. This is only
possible if a neighborhood relation is
defined on the search space.
• Every candidate solution has more than one
neighbor solution; the choice of which one
to move to is taken using only information
about the solutions in the neighborhood of
the current one, hence the name local
search.
6/15/2013

Metaheuristic
• Metaheuristic designates a computational
method that optimizes a problem by
iteratively trying to improve a candidate
solution with regard to a given measure of
quality. Metaheuristics make few or no
assumptions about the problem being
optimized and can search very large spaces
of candidate solutions.
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Background and Motivation
Metaheuristics
Integer Programming
Exact Methods
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• Local Improvement
• Tabu Search
• Iterated Local Search
• Simulated annealing
• Genetic Algorithms
• Evolutionary algorithms
• Ant Colony
Optimization
• Scatter Search
• Memetic Algorithms
• Etc….
Metaheuristics
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• Branch-and-bound
• Branch-and-cut
• Column generation
• Cutting and price
• Dynamic programming
• Lagrangian relaxation
• Linear relaxation
• Surrogate relaxation
• Lower bounds
• Etc…
Integer Programming
Exact Methods
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• Good solutions for
complex and large-scale
problems
• Short running times
• Easily adapted
Metaheuristics
Integer Programming
Exact Methods
• Proved optimal
solutions
• Important
information on
the
characteristics
and properties of
the problem.
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Metaheuristics
Integer Programming
Exact Methods
Optimized Search
Heuristics
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1



MethodsSearchdPerturbate
MethodsSearchveConstructi
MethodsSearchLocal



MethodsSearchLocalStochastic
MethodsSearchLocalticDeterminis
MethodsSearchLocal


 
MethodsSearchveCoorperati
MethodsSearchvecoorperatiNon
MethodsSearchLocal
neighborhood search, simulated annea
population-based search
It makes use the information of a set of soluti
e.g. genetic algorithms, ant algorithm, . . .Sandeep Kumar Poonia
6/15/2013

Initial
state
Partial
solution
Complete
solution
Improved
solution
Accepted
solution
Constructive procedure Iteratively improvement
procedure
Refinement:
The best solution comes from a process of repeatedly refining and
inventing alternative solutions.
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A  J  D  E  F  K  H  I  G  C  B  A
(A J D E F K H I G C B) is a complete solution.
Constructive Search Methods:
To generate a complete solution by iteratively extending partial
solutions.
A CB
J D
F E
K
I
H G
5
3
5 5
6 3
5
4107
9
6
8
3
9
4
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Perturbation Search Methods:
For a complete solution, we can easily change it into new complete
solution by modifying one or more solution components.
For example, in TSP a complete solution (ABCD) is changes into a
new solution (ADCB) by interchange the positions of B and D.
( neighborhood search methods, mutation operations i.e. )
( The Liberty Times, July 27, 2005 )
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1
For a set of complete solutions, we can easily change them into new
complete solutions by modifying one or more solution components
among the solutions. ( crossover and mutation operations in genetic
algorithms. etc. )
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Deterministic Algorithms: In each search step, it progresses
toward the complete solution by making deterministic decision.
e.g. Simplex method, Quasi-Newton algorithms, tabu search and
many other conventional algorithms.
Deterministic algorithm will produce the same solution for a given
problem instance. Even for the same instance, the stochastic
algorithm usually product distinct solutions at each run.
Ackley’s function
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4. Stochastic Algorithms:
It make a random decision at each search step. e.g. Monte Carlo
algorithms, simulated annealing, genetic algorithms, ant algorithms,
etc.
There are two cases.
(1) The available information – the objective function to be
optimized – may be considered possible erroneous or corrupted by
random noise.
(2) For the case with perfect information, we may introduce a
random element to guide us when searching for the optimum
solution.
6/15/2013

1. They are efficient for the practical uses.
2. They are simple to implement. For many applications, stochastic
algorithm is the simplest algorithm available, or fastest, or both.
3. They are very general and can be implemented for a wide class of
optimization. For example, no differential function of real valued
parameters is required. It need not be expressible in any particular
constraint language.
4. They can run in parallel. The quality of solutions may be improved
time by time.
Why stochastic algorithms?
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Transition probabilities for a deterministic algorithm:
5
3
5
7
8 9
1
3
2
1
1
1
1
1
1
1
1
1
Minimize f (x)
subject to x  S
f (xk) = 2
Configuration Graph
Deterministic algorithm will produce the same solution for a given
problem instance.
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Transition probabilities for a stochastic algorithm:
Remark: Transition probabilities may be dependent on the number
of iterations.
S8
S1
S2
S9
S7
S5
S6
S3
S4
2/4
1/4 1/8
1/8 1/4
3/4
1/3
1/3
1/3
1/8
1/8
2/41/8 1/8
3/4
1/3
1/3
1/3
3/4
1/8
1/8
1/2
1/4
1/4
1/8
1/8
1/8
2/4
1/4
Even for the same instance, the stochastic algorithm usually product
distinct solutions at each run.Sandeep Kumar Poonia
6/15/2013

S8
S1
S2
S9
S7
S5
S6
S3
S4
1 - (e-2/T)/3 - (e-1/T)/3
(e-2/T)/3 (e-1/T)/3
1/8 (e-3/T)/3
3/4
1/3
1/3
1/3
2/3 - (e-1/T)/3
1/3
1/8 1/8
3/4
1/3
1/3
1/3
3/4
1/8
1/8
1 - (e-3/T)/3 -
(e-1/T)/3
1/4
1/8
1/8
1/8
2/4
1/4
(e-1/T)/3
(e-1/T)/3
T = the number of iterations.
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There are two ways to avoid getting trapped in a local optimum.
1. Accommodate nongreedy search move. It is allowed to move to
a neighborhood state with a worse function value. ( Tabu search,
simulated annealing, . . . )
2. To increase the number of edges in the configuration graph.
However, the denser the configuration graph is, the more
inefficient search step will be.
To enlarge the neighborhood for each state.
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General local search algorithm
• G(S) is the value of the objective under
schedule S
1. Let k=1. Start with a schedule S1 and let the
best schedule S0=S1
2. Choose a schedule Sc from the neighborhood of
Sk N(Sk)
3. If Sc is accepted let Sk+1 = Sc, otherwise let
Sk+1= Sk. If G(Sk+1)<G(S0) let S0=Sk+1
4. Let k=k+1. Terminate the search if the stopping
criteria are satisfied. Otherwise return to 2.
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Local search example
Schedule representation
• Let the vector S=(j1,…,jn) represent the
schedule
– jk=j if j is the kth job in the sequence
• Use EDD to construct the initial schedule
 S1= …
• The total weighted tardiness
– Let G(S) be SwjTj under schedule S
=> G(S1)= …
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Neighborhood structure
• Manipulating S1
1. Pairwise adjacent interchange
2. Try to move a job to a different location in
the sequence
• Rules 1 and 2 above define two types of
neighborhoods N1 and N2
• N1(S1)=…
• N2(S1)=…
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Choosing Sc
• Assume we use N1
• Methods for choosing Sc from N1(Sk)
1. Randomly
2. Move the job forward that has the highest
contribution to the objective
• Follow rule 2
 Interchange jobs … and …
 Sc = ( , , , )
 G(Sc) =
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Acceptance criteria
• Is G(Sc) < G(Sk)?
• Should we consider accepting Sc if G(Sc) ≥
G(Sk) ?
• In this example we only accept if we get an
improvement in the objective
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Stopping criteria
• Max number of iterations
• No or little improvement
– We would terminate the search since we did not
improve the current schedule
• Local optimal solution
– No solution S in N(Sk) satisfies G(S)<G(Sk)
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Local search
Design criteria
i. The representation of the schedule
ii. The design of the neighborhood
iii. The search process within the
neighborhood
iv. The acceptance-rejection criteria
v. Stopping criteria
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Simulated Annealing (SA)
• Annealing: Heating of a material (metal) to
a high temperature and then cooling it at a
certain rate to achieve a desired crystalline
structure
• SA: Avoids getting stuck at a local
minimum by accepting a worse schedule Sc
with probability
P(Sk,Sc)= exp(-
G(Sc)-G(Sk)
)bk
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SA: Temperature parameter
bk ≥ 0 is the temperature (also called cooling
parameter)
• Initially the temperature is high making moves to
a worse schedule more likely
~50% chance of accepting a slightly worse schedule
seems to work well
• As the temperature decreases the probability of
accepting a worse schedule decreases
– Often, bk=Tak for some .9<a<1 and T>0
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SA algorithm
1. Set k=1 and select b1.
Select S1 and set S0=S1.
2. Select Sc (randomly) from N(Sk).
i. If G(S0)<G(Sc)<G(Sk) set Sk+1=Sc and go to 3
ii. If G(Sc)<G(S0) set S0=Sk+1=Sc and go to 3
iii. If G(Sc)>G(Sk), generate a uniform random number Uk from a
Uniform(0,1) distribution (e.g., rand() in Excel)
If Uk≤P(Sk,Sc), set Sk+1=Sc; otherwise set Sk+1=Sk.
3. Select bk+1≤ bk.
Set k=k+1.
Stop if stopping criteria are satisfied; otherwise go to
2.
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Tabu (taboo?) search
• Tabu search tries to model human memory
processes
• A “tabu-list” is maintained throughout the
search
– Moves according to the items on the list are
forbidden
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Tabu search algorithm
1. Set k=1. Select S1 and set S0=S1.
2. Select Sc from N(Sk).
i. If the move SkSc is on the tabu list set Sk+1=Sk
and go to 3
ii. If SkSc is not on the tabu list set Sk+1=Sc.
Add the reverse move to the top of the tabu list and
delete the entry on the bottom.
If G(Sc)<G(S0), set S0=Sc.
3. Set k=k+1.
Stop if stopping criteria are satisfied;
otherwise go to 2.
6/15/2013

Lecture29

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