A Modular Genetic Algorithm Specialized for Linear Constraints

Stefano Costanzo, Lorenzo Castelli, Alessandro Turco

Genetic Algorithms

Genetic Algorithms are popular stochastic optimization methods inspired by the evolutionist theory on the origin of species and natural selection.

GAs are particularly suitable for solving complex single and multi-objective problems and finding reasonably good trade-off solutions.

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How it works

GAs are designed to simulate processes in natural systems necessary for evolution, following the “Survival of the fittest“ by Charles Darwin.

GA initializes a population and improves it through iteration of the selection, genetic operators and evaluation phases.

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Genetic Algorithm Process

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Target

Effectively tackle problems with specific characteristics and maintain at least the performance of state-of-the-art Genetic Algorithms.

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Problem characteristics

• Linear constraints

• Nonlinear constraints

• Equality constraints

• Variable Bounds

• Single-objective problems

• Multi-objective problems

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Modularity

• Each phase is well defined and independent

• New valid phases are simple to design

• Multiple alternatives can co-exist

• Wide variety of specialized GA phases in literature

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Genetic Algorithm Process

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Modularity Exploitation - Selection 9

Modularity Exploitation – Genetic Operators 10

Modularity Exploitation – Before optimization 11

Before Optimization - Linear Constraints Logic

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Pre-processing

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MOGASI

Multi-Objective Genetic Algorithm for Structured Inputs

MOGASI - Complete Initialization Phase 15

MOGASI - Main Loop 16

Benchmarking

Three different categories of tests are performed:

• Constrained single-objective problem

• Unconstrained multi-objective problem

• Constrainted multi-objective problem

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Benchmarking

For each category multiple tests are chosen:

• Constrained single-objective problem

from Michalewicz Library: t01, t02, t06, t12, t13, t17, t26

• Unconstrained multi-objective problem

from NSGA-II tests: SCH, POL, KUR, ZDT1, ZDT2, ZDT4

• Constrained multi-objective problem

from NSGA-II tests: DEB, SRN, TNK, WATER

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Competitors – State of the Art GAs

• GENOCOP III

• Non-dominated Sorting Genetic Algorithm, NSGA-II

• Multi-Objective Genetic Algorithm, MOGA-II

Z. Michalewicz and G. Nazhiyath - Genocop III: co-evolutionary algorithm for numerical optimization problems with nonlinear constraints

K. Deb – A fast and elitist multiobjective genetic algorithm: NSGA-II

C. Poloni, V. Pediroda - GA coupled with computationally expensive simulations: tools to improve efficiency

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Single Objective Problems

Test name t13

Objective Function:

Constraint:

Bounds:

Average Optimal Solution

Percentage Deviation

GENOCOP 0.1422 %

MOGASI 0.0000 %

NSGA-II 43.704 %

MOGA-II 40.527 %

GENOCOP 24.9644

MOGASI 25.0000

NSGA-II 14.0738

MOGA-II 14.8680

10,00

12,00

14,00

16,00

18,00

20,00

22,00

24,00

26,00

Genocop

MOGASI

NSGA-II

MOGA-II

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Medal Table – Single-Objective Problems

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1st 2nd

3rd

4th

Multi-Objective Problems

Test Name: SRN Optimization progress with IGD:

Objective Function:

Constraint:

Bounds:

Evaluation MOGA-II NSGA-II MOGASI

1 000 0.883843 1.640582 1.094851

2 000 0.521967 0.92367 0.541842

5 000 0.305973 0.607951 0.232426

10 000 0.209635 0.531319 0.128108

15 000 0.16975 0.419232 0.092720

20 000 0.147247 0.338228 0.069383

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Medal Table - Multi-Objective Problems

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1st

2nd

3rd

Conclusions

• Problem meta-type defined by characteristics

• Exploited specific characteristics knowledge

• Kept standard GAs performance

• Good results in Benchmarks

• Easy case study expansion

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Thank you for your attention