Representation

Chapter 4, Essentials of Metaheuristics, 2013

Spring, 2014

Metaheuristics

Byung-Hyun Ha

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Outline

Introduction

Vectors

Directed encoded graphs

Trees and Genetic Programming

Lists

Rulesets

Bloat

Summary

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Introduction

Representation of individual Approach to construct, tweak, and present individual for fitness

assessment

Metaheuristics as general framework Mostly, only representation differs with regard to different problems

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Introduction

Examples of representation TSP

• Permutation-based (order-based)• e.g., 4-2-3-1 2-3-1-4 3-1-4-2 1-4-2-3

• Locus-based• The ith element represents the city following city i in the tour.• e.g., 4/3/1/2 ( 4-2-3-1, permutation-based)

• Random-key• 0.78:0.56:0.69:0.11 ( 4-2-3-1, permutation-based)

VRP (vehicle routing problems)• Using separator

• 6-9-0-2-4-7-5-0-8-1-3• Mutation and crossover?

Encoding and decoding

source: http://neo.lcc.uma.es/cEA-web/VRP.htmPhenotype Genotype

encoding

decoding

Tweak

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Introduction

Tweak in representation Phenotype (E) genotype Tweak genotype (D) phenotype Determining fitness landscape

• Example: Hamming cliff and gray coding

Remember small change!• It can help metaheuristics, usually.

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Introduction

Much of representation is an art, not a science! e.g., workflow (business process)

• How to encode and tweak?

source: http://www.tonymarston.net/php-mysql/workflow.html

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Introduction

Properties, required (Talbi, 2009) Completeness

• All solutions should be represented

Connexity• A search path must exist between any two solutions (i.e., to global optimum)

Efficiency• Easiness to manipulate

Representation-solution mapping (Talbi, 2009) One-to-one Many-to-one

• Redundancy will enlarge the size of search space.

One-to-many (indirect encoding)• A good solution should be constructed from an individual.

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Vectors

Initialization and bias Not difficult to initialize

• Some totally-random initialization method (covered already)

Bias?• e.g., solution for robot walking using heuristic (e.g., by motion capture)• But diversity is useful, particularly early on.• Some suggestions

1) Biasing is dangerous.

2) Start with values that aren’t all or exactly based on heuristic bias

Mutation Examples

• Gaussian convolution, bit-flip mutation, ...• Integer vector: Integer Randomization Mutation, Random Walk Mutation, ...

c.f., point mutation• Useful when there is less chance to get improvement by changing several

genes at a time• But, can be trapped in local optimum, e.g.,

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Vectors

Recombination One- and Two-point Crossover, Uniform Crossover Line Recombination, Intermediate Recombination ...

Heterogeneous vectors? e.g., a function with real parameters and integer parameters

Phenotype-specific mutation or crossover e.g., Jung & Moon, The Natural Crossover for the 2D Euclidean TSP,

2002 Consider fitness landscape.

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Directed Encoded Graphs

Graphs Examples

• Neural networks, finite-state automata, Petri nets, electrical circuits, ...

Types• Directed, undirected, with labels, with weights, cyclic, acyclic, recurrent, feed-

forward, sparse, dense, planar, ...• Those are constraints respecting Tweak.

Arbitrary-structured graph Our target of graph representation

Types of encoding Direct encoding

• Exact node and edge description in representation

Indirect (developmental) encoding• Some (production) rule to constructing graph, as a solution (discussed later)

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Directed Encoded Graphs

Full adjacent matrix e.g., a recurrent directed graph structure, with

• no more than 5 nodes• no more than one edge between any two node• self-edges allowed• weights for edges

Mutation examples• One vector approach

• Algorithm 45. Gaussian Convolution Respecting Zeros• Using two vectors

• One for on/off, the other for weights

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Directed Encoded Graphs

Arbitrary graph structure Initialization of graph (N, E)

• Determination of number of nodes and edges• e.g., using geometric distribution

• Creation of a node and an edge, depending on type of target graph

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Directed Encoded Graphs

Arbitrary graph structure (cont’d) Further considerations in initialization

• e.g., connected and directed acyclic graph• c.f., general algorithms textbook

Mutation• e.g., do one of the followings, random number of times

• delete a random edge• add a random edge• delete a node and all its edges• add a node• relabel a node• relabel an edge

Recombination• c.f., goal of crossover is to transfer essential and useful elements to another• Determining elements to transfer

• Selecting subset of nodes and edges, or selecting subgraph• Coping with missing target of edge and with disjoint

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Directed Encoded Graphs

Arbitrary graph structure (cont’d) Recombination (cont’d)

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Directed Encoded Graphs

Vector vs. graph representation e.g., Relocation of containers in a bay for efficient loading

• Solution as a list of movements• e.g., (1-2), (3-2), (4-5), (6-5), (4-7), (4-6)• Weakness?

• Solution as a graph

a

b

c

d e

f

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Trees and Genetic Programming

Genetic Programming How to use stochastic methods to search for and optimize small comput

er programs or other computational devices Concept of suboptimality, required

• Not simply right or wrong

Examples• Team soccer robot behavior, fitting math. equation to data set, finding finite-st

ate automata which matching given language

Representation Lists or trees, usually

• e.g., an artificial ant, sin(cos(x – sin x) + xx) for symbolic regression

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Trees and Genetic Programming

Primitives in representation Basic functions (e.g., kick-toward-goal) or CPU operations (e.g., +) Constraints of context

• e.g., 4 + kick-toward-goal(), no sense• e.g., matrix-multiply, expecting exactly two children and ...

Tweaks need to maintain closure (valid individuals)

Fitness assessment Conversion data (genotype) to code (phenotype), and evaluate Examples

• Symbolic regression: sum of squared errors• Artificial ant: amount of food eaten

Tree-Style Genetic Programming Pipeline Sec. 3.3.3 One of popular algorithm for Genetic Programming (but not limited to)

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Trees and Genetic Programming

Initialization New trees by repeatedly selecting from a function set

• Considering arity (predefined number of children)• e.g., Grow, Full, Ramped Half-and-Half, PTC2 algorithms

Ephemeral random constants• Handling constants for leaves (e.g., 0.2462, 0.9, –2.34, 3.14, “s%&e:m”)• Special leaf nodes to be transformed into randomly-generated constant

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Trees and Genetic Programming

Recombination e.g., subtree crossover: swap two selected subtrees

• Non-homologous (i.e., highly mutative)

homologous: individual crossing over with itself will make copies of itself

Mutation Examples

• Subtree mutation: replacing random subtree with randomly-generated one• Replacing random non-leaf node with one of its subtrees• Picking random non-leaf node and swapping its subtrees• Mutating ephemeral random constants by introducing some noise• Swapping two disjoint subtrees

c.f., not popular because usually crossover is non-homologous

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Trees and Genetic Programming

Forests e.g., forest of soccer robot team with each member as tree

Automatically defined functions (ADF) Not predefined functions but trees called by primary tree c.f., Modularity

• In case that we believe a good solution has repetitive part

Strongly-Typed Genetic Programming

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Trees and Genetic Programming

Cellular encoding Indirect encoding (developmental encoding)

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Lists

Grammatical Evolution: using predefined grammar for tree Trees generated by lists (indirect encoding)

• c.f., http://en.wikipedia.org/wiki/Backus-Naur_form

Pros and cons• Almost always valid tree, reduced size of search space• Tiny changes early in list result in gigantic changes (un-smoothness).

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Rulesets

A policy as solution of problem Consisting of a set of rules e.g., stock trading program, entities in simulations

State-action rules Typical form

• a b ... y z• e.g., (left sonar value > 3.2) (forward sonar value 5.0) (turn left to 50)

An interpretation• Mapping from state space into actions

Under-specification and over-specification• Default rules, vote, ...

Fitness assessment• On a ruleset, or on a series of rules

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Rulesets

Production rules Typical form

• a b c ... z Modular indirect encoding

• Describing large complex solution with lots of repetitions by small and compact rule (search) space

e.g., 8-node directed unlabeled graph structure as solution

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Rulesets

Production rules (cont’d) e.g., Lindenmayer systems (L-systems)

• e.g., Koch Curve• F F + F – F – F + F• F: draw a line forward, +: turn left, –: turn right

F

F+F-F-F+F

F+F-F-F+F+F+F-F-F+F-F+F-F-F+F-F+F-F-F+F+F+F-F-F+F

F+F-F-F+F+F+F-F-F+F-F+F-F-F+F-F+F-F-F+F+F+F-F-F+F+F+F-F-F+F+F+F-F-F+F-F+F-F-F+F-F+F-F-F+F+F+F-F-F+F-F+F-F-F+F+F+F-F-F+F-F+F-F-F+F-F+F-F-F+F+F+F-F-F+F-F+F-F-F+F+F+F-F-F+F-F+F-F-F+F-F+F-F-F+F+F+F-F-F+F+F+F-F-F+F+F+F-F-F+F-F+F-F-F+F-F+F-F-F+F+F+F-F-F+F

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Bloat

Code bloat or code growth A problem with variable-sized representation Far from optimum usually, memory consumption, ... and ugly

Common ways of handling Limiting size when individual is Tweaked Editing individual, to remove introns and the like Punishing individual for being very large

• e.g., linear parsimony pressure (problem?)• revised fitness f = r – (1 – )s, where r: fitness, s: size of individual

• e.g., non-parametric parsimony pressure

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Summary

Phenotype & genotype

Encoding & decoding

Representations Vectors Graphs

+ Indirect-encoded graphs (edge encoding)

Trees+ Indirect-encoded trees (Grammatical Evolution)

Lists Rulesets

Bloat