Study on Paretofront of Multiobjective Optimization Using Immune Algorithm

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Study on Pareto-front of Multi-objective
Optimization Using Immune Algorithm

• Guang xing Tan and Zong Yuan Mao
• Science and Engineering, South China University
  of Technology
• 4th IEEE Conf. on Machine Learning and
  Cybernetics, 2006

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Introduction

• Authors propose an immunologically-inspired
  algorithm that finds pareto optimal solutions of
  multi-objective optimization problem
• The algorithm is designed based on immune network
  metaphor
• Antibodies stimulate and suppress each other in
  antibody network
• It does not use non-domination ranking scheme
• Fitness ranking based on Pareto fitting ratio is
  used
• Claim It is difficult for the algorithm based on
  non-domination ranking scheme to solve MOP which
  has a more than three objectives
• Its solutions are greatly influenced by the
  dimension of objective space
• Especially, the time complexity of algorithm
  increases as dimension increases
• Goal To simplify the comparison mechanism for
  fitness ranking procedure
• Reduce the time complexity of the algorithm with
  high-dimensional objective optimization problem

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Definition

• Multiple optimization problem
• Pareto optimal set
• Pareto front

S search spacen-dim rectangle defined bylower
and upper boundsof a variable
M of candidate solutions i.e. size of
population
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Definition

• An variable Xstar is said to be feasible Pareto
  optimal
• if there exists no other variable Xbar such that
• Pareto fitting ratio

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• Theorem
• xj is a feasible Pareto optimal iff PFRf(xj) gt
  1
• (proof)
• If there is a xj which dominates xj
• ? PFRf(xj) is less than 1
• ? Theorem guarantees that population contains
  individuals that are not dominated to each other
  and to all individuals

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The proposed mechanism

• Step 1 recognition of antigen
• Identification of the optimization problem
• Step 2 generation of antibodies
• Randomly create N vectors, each of which
  representsa candidate solution
• Step 3 reaction of antibodies
• While (stopping criteria is not met) Do
• Calculate Fitness value based on PFR of each
  antibody

Cg
N
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The proposed mechanism (2)

• Step 4 memory of past infections are maintained
• M-highest-fitness antibody (whose fitness(xi) gt
  0) will be selected and sorted into a memory
  antibody set Mp
• Step 5 antibody stimulation and mutation
• Perform clone and mutation on individuals in Mp
• (fitness proportional mutation scheme is used)
• fitness(i) is large ? small
  move fitness(i) is small ? large move

Mp
M
Mpc
M
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The proposed mechanism (3)

• Step 6 antibody suppression
• Eliminate antibodies whose fitness is less than
  sigma
• Sigma suppression threshold
• Step 7 generation of new antibody
• Concatenate Mpc into the population
• Create new Nxd antibodies randomly andadd them
  into the population
• Go back to Step 3

Cg
C
C
Cg1
Mpc
new
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Evaluation

• Evaluate validity of the proposed mechanism
• Population size N20, lambda0.02, sigma0.015
• Run 10 iterations of 5 generations

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• Convex problem and non-convex problem

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Comparison

• MISA (Coello Coello and Cortes, 2002) may be the
  first attempt to solve general multiobjective
  optimization problems using artificial immune
  systems.
• MISA encodes the input variables of the problem
  to be solved by binary strings
• It finds non-dominated and feasible solutions in
  the population
• It performs mutation to their clones
• MISA updates its external population by using the
  grid based techniques used in PAES (Knowles and
  Corne, 2000).

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• Coverage of two sets
• DTLZ1, 2

1 Ic(IA, MISA) 2 Ic(MISA, IA)
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Running time

• Running time over different of objectives
• Finding good individuals
• Domination ranking calculation (MISA)
• PFR calculation (the proposed mechanism)
• Mutation
• Update population elimination, insertion,
  concatenation

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Conclusion

• Authors propose immunologically-inspired
  algorithm to find Pareto optimal solutions for
  MOP
• They propose Pareto fitting ratio scheme for
  finding non-dominating individuals
• Compared to another AIS based mechanism, the
  proposed mechanism reduce running time

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• Coverage of two sets
• DTLZ1, 2