Optimization with Genetic Algorithms

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# Optimization with Genetic Algorithms - PowerPoint PPT Presentation

Optimization with Genetic Algorithms. Walter Reade October 31, 2002 TAG Meeting. Outline . Background on Optimization Introduction to Genetic Algorithms Using GAs to Solve Difficult Problems A MatLab Implementation Summary / Questions. How Do We Find the Minimum?.

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### Optimization withGenetic Algorithms

October 31, 2002

TAG Meeting

Outline
• Background on Optimization
• Introduction to Genetic Algorithms
• Using GAs to Solve Difficult Problems
• A MatLab Implementation
• Summary / Questions
• Move in the direction of steepest gradient.
• Simple to implement, guaranteed convergence.
• Must know something about the derivative.
• Can easily get stuck in a local minimum.
Stochastic Methods
• Heuristic
• Using “Rules of Thumb”
• Metaheuristic
• A framework of heuristics used to update a set of solutions during a search.
• Simulated Annealing
• Tabu Search
• Ant Colony Systems
Genetic Algorithms
• Use a population of possible solutions to the search space.
• Each solution is encoded in a string called a chromosome (or genome).
• Chromosomes are evaluated for fitness each generation (iteration); chromosomes that are more fit have a high probability of surviving.
Genetic Algorithms (cont.)
• Once the surviving population is chosen, different “parent” chromosomes are combined to form “child” chromosomes.
• Chromosomes may undergo mutation.
• A new generation is formed, the process is repeated.
• By selection, cross-over, and mutation, GAs search the solution space while creating stronger solutions over each generation.
Fitness and Selection
• Roulette Wheel
• Competition
• Etc.
Cross-Over
• Replaces two parent solutions with two children solutions.
• Mechanism for covering large area of search space.
Mutation
• Operates on a single chromosome.
• Mechanism to improve local search space.
Advantages to using GAs
• Flexible and adaptive to a wide variety of problems.
• Robust, global search capability.
• Does not require the solution space to be smooth, continuous, or differentiable.
• Can be used in permutation problems.
• No practical drawbacks.
• Slow local convergence
• Perceived learning overhead
Function optimization

Job shop scheduling

Process planning

Assembly line optim.

Process control

Airplane landing

Nested design

Keyboard layout

Creativity

VLSI

Traveling Sales Man

Chemical kinetics

Etc.

Etc.

Etc.

Applications
Difficult Problems
• Appeared in Jan/Feb 2002 SIAM News in the 100-Dollar, 100-Digit Challenge.
• exp(sin(50*x)) + sin(60*exp(y)) + sin(70*sin(x)) + sin(sin(80*y)) - sin(10*(x+y)) + 0.25*(x^2 + y^2)
• The genetic algorithm was able to solve this to 10 digits of precision in 2000 generations, which took 25 seconds on a P-III 1.0 GHz. (35% success rate)
Permutation (Order-based) Problems
• Time-share Example
• One condo building at a ski resort
• Four identical condo units
• 16 week ski season – 64 total owners
• 2nd choice = 2 free lift tickets per person, 3rd choice = 5 free tickets, otherwise \$\$ and 7 free tickets.
• 5 out of 16 weeks are twice as popular
• Maximum occupancy = 22
• Possible solutions: 1x1067
Results of GA
• A previous published result (using SA) found a minimum of 224 after 261 iterations, and no improvement after 1,000,000 iterations.
• The GA found a cost of 200 after 2,150 iterations, and a minimum of 172 after 250,000 iterations.
• (Author of previous work was “astonished” at the new result.)
Using GAs in MatLab
• http://www.ie.ncsu.edu/mirage/GAToolBox/gaot/
% Bounds on the variables

bounds = [-5 5; -5 5];

% Evaluation Function

evalFn = 'Four_Eval';

evalOps = [];

% Generate an intialize population of size 80

startPop=initializega(80,bounds,evalFn,[1e-10 1]);

% GA Options [epsilon float/binary display]

gaOpts=[1e-10 1 0];

% Termination Operators -- 500 Generations

termFns = 'maxGenTerm';

termOps = [500];

% Selection Function

selectFn = 'normGeomSelect';

selectOps = [0.08];

% Crossover Operators

xFns = 'arithXover heuristicXover simpleXover';

xOpts = [1 0; 1 3; 1 0];

% Mutation Operators

mFns = 'boundaryMutation multiNonUnifMutation nonUnifMutation unifMutation';

mOpts = [2 0 0;3 200 3;2 200 3;2 0 0];

% Apply the genetic algorithm

[soln endPop bestPop trace]=ga(bounds,evalFn,evalOps,startPop,gaOpts,termFns,termOps,selectFn,selectOps,xFns,xOpts,mFns,mOpts);

MatLab Code
Evaluation Function

function [x, soln] = Four_Eval(x, options)

soln = -(exp(sin(50*x(1))) + sin(60*exp(x(2))) + sin(70*sin(x(1))) + ...

sin(sin(80*x(2))) - sin(10*(x(1)+x(2))) + 0.25*(x(1)^2 + x(2)^2));

Time Share Evaluation Function

function [assignment, soln] = local_min(assignment, options)

global family_info

cost = 0;

occupancy = zeros(16,1);

for i = 1:64

if ceil(assignment(i)/4) == family_info(i,2) % first choice

cost = cost + 0;

elseif ceil(assignment(i)/4) == family_info(i,3) % second choice

cost = cost + 2*family_info(i,1);

elseif ceil(assignment(i)/4) == family_info(i,4) % third choice

cost = cost + 4*family_info(i,1);

else % didn't get any choice

cost = cost + 50 + 7*family_info(i,1);

end

building = ceil(assignment(i)/4);

occupancy(building) = occupancy(building) + family_info(i,1);

end

for i = 1:16

if occupancy(building) > 22

cost = cost + 1000;

end

end

soln = -cost;

Summary
• Genetic Algorithms are:
• Powerful
• Flexible
• Easy to use and understand
• Consider using a GA for your next optimization problem!