ECE 103 Engineering ProgrammingChapter 52

Generic Algorithm

Herbert G. Mayer, PSU CSStatus 6/4/2014

Initial content copied verbatim fromECE 103 material developed by

Professor Phillip Wong @ PSU ECE

Syllabus Generic Algorithm C C C c

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Genetic Algorithms

A genetic algorithm (GA) can solve certain classes of optimization problems in which: The details of the problem are not well known. The search space is large.

Genetic algorithms are based on the biological principles of evolution and selection.

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General Plan:

An initial randomized population of candidate solutions is constructed.

For each subsequent generation: The current generation is evaluated to determine its

“fitness”. The most fit candidates are selected for

“reproduction”. A new generation of offspring is created.

This continues until a suitable solution is found or a fixed number of generations is reached.

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Solutions are encoded as “bit strings”, e.g., a sequence of 1’s and 0’s stored in an array.

Fitness is determined by how well a candidate solution meets a pre-defined criterion.

Roulette wheel selection:An individual’s probability of being selected is directly proportional to its fitness.

Only selected individuals are allowed to reproduce.

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Typical reproduction methods: Crossover - potentially copies good (or bad) traits:

Mutation (helps introduce genetic variety):

Parent 1

1011010 010100110

Parent 2

0011010 110110101

Child 1

1011010 110110101

Child 2

0011010 010100110

Before 1101101001101110

After 1101100001101110

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Expressed in terms of arrays:

Create initial population Pcurrent from N randomized bit strings (arrays)Create blank population Pnext to hold N bit strings (arrays)

FOR up to G generations

Evaluate population Pcurrent for fitness

Initialize roulette wheel using fitness results

Select parents from Pcurrent using roulette wheel probabilities

Create population Pnext by generating N offspring using genetic operators

Copy Pnext to Pcurrent

END FOR

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Depending on N, L, and # of generations to run, the array copy operation can be expensive.

Why not use pointers to arrays instead?

Pcurrent

N bit-stringsof length L

(NL array)

Evaluatefitness

Initialize roulette wheel

Roulette wheelselects parents

Generate offspring

Pnext

N bit-stringsof length L

(NL array)

Copy the contents of array Pnext to array Pcurrent one element at a time.

In other words, the next generation becomes the current generation.

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Expressed in terms of pointers:

Create initial population PA from N randomized bit strings (arrays)Create blank population PB to hold N bit strings (arrays)

Assign pointer Pcurrent → PA and pointer Pnext → PB

FOR up to G generations

Evaluate population Pcurrent for fitness

Initialize roulette wheel using fitness results

Select parents from Pcurrent using roulette wheel probabilities

Create population Pnext by generating N offspring using genetic operators

Swap pointers Pcurrent and Pnext

END FOR

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Gen k

Pcurrent PA

N bit-stringsof length L

(NL array)

Evaluatefitness

Initialize roulette wheel

Roulette wheelselects parents

Generate offspring

PB

N bit-stringsof length L

(NL array)

Pnext

Swap the pointers Pcurrent and Pnext.

Gen k+1

Pcurrent PB

N bit-stringsof length L

(NL array)

Evaluatefitness

Initialize roulette wheel

Roulette wheelselects parents

Generate offspring

PA

N bit-stringsof length L

(NL array)

Pnext

Swap the pointers Pcurrent and Pnext.

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Example:Find the minimum of the following equation:f (x, y) = 4x2 – 2.1x4 + (1/3)x6 + xy – 4y2 + 4y4

over the range –5 x, y 5

Using Mathcad, the global minimum value forf (x, y) on the defined interval is –1.0316285.

This occurs at:(x0, y0) = (– 0.089842, +0.7126564)

(x1, y1) = ( +0.089842, –0.7126564)

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Run a genetic algorithm simulation using these reproduction methods:

(GA-1) M1 = 1-point crossover and M2 = complement randomly chosen bit position. Probabilities are p1 = 0.97 and p2 = 0.03. The GA is not elitist.(GA-2) Identical to GA-1 except that M1 = 2-point crossover.(GA-3) M1 = 1-point crossover with p1 = 1.0. No M2 is defined. Each bit in each member of the offspring population is complemented with a probability of 10-5. The GA is not elitist.(GA-4) Identical to GA-3 except that the GA is elitist.

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Simulation results:%Relative Error in Minimum Value & (x, y) for Best and Worst Runs

@ 200th Generation (N=50, L=28, Runs=25)Best Run Worst Run

RelErr% x y RelErr% x yGA-1 0.000 -0.090643 0.712629 6.864 -0.086980 0.612525GA-2 0.000 -0.089422 0.713239 7.773 -0.002747 -0.625954GA-3 0.013 -0.086980 0.708967 13.659 -0.098578 -0.713239GA-4 0.075 -0.089422 0.702863 15.357 -0.000916 -0.823109

Comparison to Mathcad locations for minimum:(x0, y0) = (– 0.089842, +0.7126564)(x1, y1) = ( +0.089842, –0.7126564)

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0

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Example:NASA experiment to design an X-band antenna(8 to 12 GHz range for satellite communications):

Result: improved performance and efficiency