GENETIC ALGORITHMS Tanmay, Abhijit, Ameya, Saurabh

Inspiration - Evolution Natural Selection: – “Survival of the Fittest” – favourable traits become common and unfavourable traits become uncommon in successive generations Sexual Reproduction: – Chromosomal crossover and genetic recombination – population is genetically variable – adaptive evolution is facilitated – unfavourable mutations are eliminated

Overview  Inspiration  The basic algorithm  Encoding  Selection  Crossover  Mutation  Why Genetic Algorithms work ?  Schemas  Hyper-planes  Schema Theorem  Strengths and Weakness  Applications  TSP  Conclusion

THE BASIC ALGORITHM Ameya Muley

Encoding of Solution Space  Represent solution space by strings of fixed length over some alphabet  TSP:  ordering of points  Knapsack:  inclusion in knapsack ADBECBEDAC B AC ED

Selection Fitness function: – f(x), x is a chromosome in the solution space – f(x) may be: an well-defined objective function to be optimised – e.g. TSP and knapsack a heuristic – e.g. N-Queens Probability distribution for selection: Fitness proportional selection

Operators-Crossover and Mutation Crossover: – Applied with high probability – Position for crossover on the two parent chromosomes randomly selected – Offspring share characteristics of well-performing parents – Combinations of well-performing characteristics generated Mutation: – Applied with low probability – Bit for mutation randomly selected – New characteristics introduced into the population – Prevents algorithm from getting trapped into a local optimum

The Basic Algorithm 1. Fix population size M 2. Randomly generate M strings in the solution space 3. Observe the fitness of each chromosome 4. Repeat: 1. Select two fittest strings to reproduce 2. Apply crossover with high probability to produce offspring 3. Apply mutation to parent or offspring with low probability 4. Observe the fitness of each new string 5. Replace weakest strings of the population with the offspring until i. fixed number of iterations completed, OR ii. average/best fitness above a threshold, OR iii. average/best fitness value unchanged for a fixed number of consecutive iterations

Example Problem specification: – string of length 4 – two 0’s and two 1’s – 0’s to the right of the 1’s Solution space: Fitness function (heuristic): – f(x) = number of bits that match the ones in the solution Initialization (M = 4):

Example (contd.)  After iteration 1:  After iteration 2:

WHY GENETIC ALGORITHMS WORK? Tanmay Khirwadkar

Schemas  Population  Strings over alphabet {0,1} of length L  E.g.  Schema  A schema is a subset of the space of all possible individuals for which all the genes match the template for schema H.  Strings over alphabet {0,1,*} of length L  E.g.

Hyper-plane model  Search space  A hyper-cube in L dimensional space  Individuals  Vertices of hyper-cube  Schemas  Hyper-planes formed by vertices 0**

Sampling Hyper-planes  Look for hyper-planes (schemas) with good fitness value instead of vertices (individuals) to reduce search space  Each vertex  Member of 3 L hyper-planes  Samples hyper-planes  Average Fitness of a hyper-plane can be estimated by sampling fitness of members in population  Selection retains hyper-planes with good estimated fitness values and discards others

Schema Theorem  Schema Order O(H)  Schema order, O(.), is the number of non ‘*’ genes in schema H.  E.g. O(1**1*) = 2  Schema Defining Length δ (H)  Schema Defining Length, δ (H), is the distance between first and last non ‘*’ gene in schema H  E.g. δ (1**1*) = 4 – 1 = 3  Schemas with short defining length, low order with fitness above average population are favored by GAs

Formal Statement  Selection probability  Crossover probability  Mutation probability  Expected number of members of a schema

Why crossover and mutation?  Crossover  Produces new solutions while ‘remembering’ the characteristics of old solutions  Partially preserves distribution of strings across schemas  Mutation  Randomly generates new solutions which cannot be produced from existing population  Avoids local optimum

STRENGTHS AND WEAKNESS Abhijit Bhole

Area of application GAs can be used when:  Non-analytical problems.  Non-linear models.  Uncertainty.  Large state spaces.

Non-analytical problems  Fitness functions may not be expressed analytically always.  Domain specific knowledge may not be computable from fitness function.  Scarce domain knowledge to guide the search.

Non-linear models  Solutions depend on starting values.  Non – linear models may converge to local optimum.  Impose conditions on fitness functions such as convexity, etc.  May require the problem to be approximated to fit the non-linear model.

Uncertainty  Noisy / approximated fitness functions.  Changing parameters.  Changing fitness functions.  Why do GAs work? Because uncertainty is common in nature.

Large state spaces  Heuristics focus only on the immediate area of initial solutions.  State-explosion problem: number of states huge or even infinite! Too large to be handled.  State space may not be completely understood.

Characteristics of GAs  Simple, Powerful, Adaptive, Parallel  Guarantee global optimum solutions.  Give solutions of un-approximated form of problem.  Finer granularity of search spaces.

When not to use GA!  Constrained mathematical optimization problems especially when there are few solutions.  Constraints are difficult to incorporate into a GA.  Guided domain search is possible and efficient.

PRACTICAL EXAMPLE - TSP Saurabh Chakradeo

TSP Description  Problem Statement: Given a complete weighted undirected graph, find the shortest Hamiltonian cycle. (n nodes)  The size of the solution space in (n-1)!/2  Dynamic Programming gives us a solution in time O(n 2 2 n )  TSP is NP Complete

TSP Encoding  Binary representation  Tour is represented as ( )  Path representation  Natural – ( )  Adjacency representation  Tour is represented as ( )  Ordinal representation  A reference list is used. Let that be ( ).  Tour is represented as ( )

TSP – Crossover operator  Order Based crossover (OX2)  Selects at random several positions in the parent tour  Imposes the order of nodes in selected positions of one parent on the other parent  Parents: ( ) and ( )  Selected positions, 2 nd, 3 rd and 6 th  Impose order on ( ) &( )  Children are ( ) and ( )

TSP – Mutation Operators  Exchange Mutation Operator (EM)  Randomly select two nodes and interchange their positions.  ( ) can become ( )  Displacement Mutation Operator (DM)  Select a random sub-tour, remove and insert it in a different location.  ( 1 2 [3 4 5] 6 ) becomes ( )

Conclusions  Plethora of applications  Molecular biology, scheduling, cryptography, parameter optimization  General algorithmic model applicable to a large variety of classes of problems  Another in the list of algorithms inspired by biological processes – scope for more parallels?  Philosophical Implication:  Are humans actually moving towards their global optimum?

References  Adaptation in Natural and Artificial Systems, John H. Holland, MIT Press,  Goldberg, D. E Genetic Algorithms in Search, Optimization and Machine Learning. 1st. Addison- Wesley Longman Publishing Co., Inc.  Genetic Algorithms for the Travelling Salesman Problem: A Review of Representations and Operators, P. Larranaga et al., University of Basque, Spain. Artificial Intelligence Review, Volume 13, Number 2 / April, 1999