Evolutionary Design (2) Boris Burdiliak

Topics Representation Representation Multiple objectives Multiple objectives

Representation – Introduction EA are based on natural evolution EA are based on natural evolution evolution provides a quick and easy way to solve difficult problems (scheduling, machine learning, ordering problems, data mining, control, design optimisation) evolution provides a quick and easy way to solve difficult problems (scheduling, machine learning, ordering problems, data mining, control, design optimisation)

Representation – Problems traditional implementations of evolutionary search suffer from the same drawbacks as all conventional search algorithms traditional implementations of evolutionary search suffer from the same drawbacks as all conventional search algorithms they rely on a good parameterization to find a good solution they rely on a good parameterization to find a good solution evolution is limited by the representation it can modify evolution is limited by the representation it can modify

Explorative evolution evolution can search for good search spaces, even as it searches within a space evolution can search for good search spaces, even as it searches within a space dimensionality of a space, the parameterization, the representation of solutions dimensionality of a space, the parameterization, the representation of solutions solutions = inventions, not improvements solutions = inventions, not improvements

Exploring the Explorer what is evolution doing when it explores/optimises? what is evolution doing when it explores/optimises? fixed-length parameterization fixed-length parameterization evolution = optimisation evolution = optimisation parameters define a set of components parameters define a set of components explores new ways of constructing a solution explores new ways of constructing a solution

A Framework for Expl. Evolution EA EA GA/GP (genotype/phenotype distinction) EA/ES (no distinction) GA/GP (genotype/phenotype distinction) EA/ES (no distinction) prob.: multiple objectives, premature convergence, changing fitness functions prob.: multiple objectives, premature convergence, changing fitness functions Genotype Representation Genotype Representation defines the search space(defines components) defines the search space(defines components) prob.: dissimilar problems are close to e.o., disruption of inheritance prob.: dissimilar problems are close to e.o., disruption of inheritance

A Framework for Expl. Evolution 2 Embryogeny Embryogeny mapping process from genotype to phenotype mapping process from genotype to phenotype provide the mechanism for constructing whole solutions from components provide the mechanism for constructing whole solutions from components external: programmer writes the software that performs the mapping external: programmer writes the software that performs the mapping explicit: every step of growth process explicitly held as a part of genotype, and evolved explicit: every step of growth process explicitly held as a part of genotype, and evolved implicit: set of rules (encoded as bin. strings in genotype) implicit: set of rules (encoded as bin. strings in genotype) Phenotype Representations Phenotype Representations allow direct evaluation of fitness function allow direct evaluation of fitness function Fitness functions Fitness functions provide evaluation score for every solution provide evaluation score for every solution

Multi objectives optimise function F(x) under additional constraints, i.e. optimise function F(x) under additional constraints, i.e. max {x} F(x) max {x} F(x) g_1(x,p) <= 0, … g_l(x,p) <= 0 g_1(x,p) <= 0, … g_l(x,p) <= 0 p(p_1,…,p_u) – real-valued parameters p(p_1,…,p_u) – real-valued parameters relative importance, interactions of objectives (also contradictory) relative importance, interactions of objectives (also contradictory)

Pareto optimisation Def: We will say that a point x  D Pareto- dominates a point y  D with respect to function F, denoted y  x, if  {i=1..k} (f_i(y)  f_i(x)) and at least one of inequalities is strict Def: We will say that a point x  D Pareto- dominates a point y  D with respect to function F, denoted y  x, if  {i=1..k} (f_i(y)  f_i(x)) and at least one of inequalities is strict We say that point x_p  D is Pareto-optimal or non-dominated (for a given function F) if there is no point y  D that Pareto-dominates x. We say that point x_p  D is Pareto-optimal or non-dominated (for a given function F) if there is no point y  D that Pareto-dominates x. Set F  D is called Pareto front with respect to function F if every element x F is Pareto optimal with respect to function F Set F  D is called Pareto front with respect to function F if every element x F is Pareto optimal with respect to function F

Selection phase not all elements from the Pareto front are of a good fitness value not all elements from the Pareto front are of a good fitness value Pareto tournament Pareto tournament standard tournament selection (best individual according to Pareto ordering) standard tournament selection (best individual according to Pareto ordering) Pareto sort Pareto sort sort according to Pareto ordering sort according to Pareto ordering if both ind. are dominant, sort acc. to fitness value if both ind. are dominant, sort acc. to fitness value Pareto truncation Pareto truncation choose parent only among the non-dominated choose parent only among the non-dominated

#of non-dominated elements population size = 100 population size = 100 standard tournament (size 2) standard tournament (size 2) averaged over 50 runs averaged over 50 runs optimising 3 parameters optimising 3 parameters non-dominated elements non-dominated elements after 50 runs: cca 42% after 50 runs: cca 42% after 150 runs: >50% after 150 runs: >50% after 350 runs: almost 70% after 350 runs: almost 70% after 500 runs: cca 85% after 500 runs: cca 85% Pareto front is too large Pareto front is too large

Pareto ranking Def: Pareto rank r in a set X={x_1,…,x_n} is assigned in a following way Def: Pareto rank r in a set X={x_1,…,x_n} is assigned in a following way (x  X)  r(x) 1 (x  X)  r(x) 1 (  x  {x_1,…x_n}) (  y  {x_1,…,x_n} \ {x}) (  x  {x_1,…x_n}) (  y  {x_1,…,x_n} \ {x}) if (r(x)=r(y)  x > y) r(x)  r(x)+1 if (r(x)=r(y)  x > y) r(x)  r(x)+1 if (r(x)=r(y)  x < y) r(y)  r(y)+1 if (r(x)=r(y)  x < y) r(y)  r(y)+1

Conclusions requirement is for methods that can work with a qualitative in addition to a quantitative characterization of importance requirement is for methods that can work with a qualitative in addition to a quantitative characterization of importance preference methods preference methods fuzzy set methods fuzzy set methods agent based systems which model human- based solution evaluation process agent based systems which model human- based solution evaluation process

Fuzzy preferences transforming qualitative relationships between objectives in multi-objectives optimization into quantitative attributes transforming qualitative relationships between objectives in multi-objectives optimization into quantitative attributes characterisations for every 2 objectives: characterisations for every 2 objectives: much less important << much less important << less important < less important < equally important == equally important == don’t care # don’t care # more important > more important > much more important >> much more important >>

Don’t care relation don’t care is treated exactly as equally important don’t care is treated exactly as equally important hesitations hesitations weak preference weak preference can not decide: a > b and a ==b, being sure that not (b > a) can not decide: a > b and a ==b, being sure that not (b > a) incomparability incomparability can not decide: a > b and b > a can not decide: a > b and b > a

Method used k objectives – k(k-1)/2 questions -> construct the fuzzy preference relation R k objectives – k(k-1)/2 questions -> construct the fuzzy preference relation R compute complete oredering on A using the relation graph G=(A,R) compute complete oredering on A using the relation graph G=(A,R) define relations: define relations: == equally important == equally important < is less important < is less important << is much less important << is much less important ~ not important ~ not important ! important ! important

Algorithm set of objectives O set of objectives O construct the equivalence classes C_i (relation ==), choose one element x_i from each class X={x_1,…,x_m} construct the equivalence classes C_i (relation ==), choose one element x_i from each class X={x_1,…,x_m} … for each xi compute weight w(xi) for each xi compute weight w(xi)

Weighted Pareto method Definition of non-dominated vector Definition of non-dominated vector Definition of w-non-dominanace Definition of w-non-dominanace Definition of (w, tau)-non-dominance Definition of (w, tau)-non-dominance

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