Evolutionary Computing. Constraint Handling. Bu-Ali Sina University Computer Engineering Dep. Fall 2010

Transcription

1 Evolutionary Computing Constraint Handling Bu-Ali Sina University Computer Engineering Dep. Fall 2010

2 Overview Motivation and the trouble What is a constrained problem? Evolutionary constraint handling A selection of related work Conclusions, observations, and suggestions

3 Motivation Why bother about constraints? Practical relevance: a great deal of practical problems are constrained. Theoretical challenge: a great deal of untractable problems (NP-hard etc.) are constrained. Why try with evolutionary algorithms? EAs show a good ratio of (implementation) effort/performance. EAs are acknowledged as good solvers for tough problems.

4 What is a constrained problem? Consider the Traveling Salesman Problem for n cities, C = {city 1,, city n } If we define the search space as S = C n, then we need a constraint requiring uniqueness of each city in an element of S S = {permutations of C}, then we need no additional constraint. The notion constrained problem depends on what we take as search space

5 What is constrained search? What is free search? Free search: standard mutation and crossover preserve membership of S, i.e., mut(x) S and cross(x,y) S The notion free search depends on what we take as standard mutation and crossover. mut is standard mutation if for all x 1,, x n, if mut( x1,, xn ) = x 1,, x n, then x i domain(i) cross is standard crossover if for all x1,, xn, y1,, yn, if cross( x1,, xn, y1,, yn ) = z1,, zn, then zi {xi, yi} discrete case zi [xi, yi] continuous case

6 Free search space Free search space: S = D 1 D n one assumption on D i : if it is continuous, it is convex the restriction s i D i is not a constraint, it is the definition of the domain of the i-th variable membership of S is coordinate-wise, hence a free search space allows free search A problem can be defined through An objective function (to be optimized) Constraints (to be satisfied)

7 Types of problems Objective function Constraints yes no yes no Constrained optimization problem Free optimization problem Constraint satisfaction problem No problem

8 Free Optimization Problems Free Optimization Problem: S, f, S is a free search space f is a (real valued) objective function on S Solution of an FOP: s S such that f (s) is optimal in S FOPs are easy, in the sense that: it's only optimizing, no constraints and EAs have a basic instinct for optimization

9 Example of an FOP Ackley function f ( x) n n 1 = x exp 0.2 i xi + + n exp cos(2π ) 20 i= 1 n i= 1 e

10 Constraint Satisfaction Problems Constraint Satisfaction Problem: S,, Φ S is a free search space Φ is a formula (Boolean function on S) Φ is the feasibility condition S Φ = {s S Φ(s) = true} is the feasible search space Solution of a CSP: s S such that Φ(s) = true (s is feasible)

11 Constraint Satisfaction Problems Φ is typically given by a set (conjunction) of constraints c i = c i (x j1,, x jni ), where n i is the arity of c i c i D j1 D jni is also a common notation FACTS: the general CSP is NP-complete every CSP is equivalent with a binary CSP, where all n i 2 Constraint density and constraint tightness are parameters that determine how hard an instance is

12 Example of a CSP Graph 3-coloring problem: G = (N, E), E N N, N = n S = D n, D = {1, 2, 3} Φ(s) = Λ e E c e (s), where c e (s) = true iff e = (k, l) and s k s l

13 Constrained optimization problems Constrained Optimization Problem: S, f, Φ S is a free search space f is a (real valued) objective function on S Φ is a formula (Boolean function on S) Solution of a COP: s S Φ such that f(s) is optimal in S Φ

14 Example of a COP Travelling salesman problem S = C n, C = {city 1,, city n } Φ(s) = true i, j {1,, n} i j s i s j n f(s) = i= 1 dist( s i, s i+ 1 ), where sn+ 1 : = s 1

15 Solving CSPs by EAs EAs need an f to optimize S,, Φ must be transformed first to a 1. FOP: S,, Φ S, f, or 2. COP: S,, Φ S, f, Ψ The transformation must be (semi-)equivalent, i.e. at least: 1. f (s) is optimal in S Φ(s) 2. ψ (s) and f (s) is optimal in S Φ(s)

16 Constraint handling 1 Constraint handling interpreted as constraint transformation Case 1: CSP FOP All constraints are handled indirectly, i.e., Φ is transformed into f and later they are solved by `simply optimizing in S, f, Case 2: CSP COP Some constraints handled indirectly (those transformed into f) Some constraints handled directly (those remaining constraints in ψ) In the latter case we also have constraint handling in the sense of treated during the evolutionary search

17 Constraint handling Constraint handling has two meanings: 1. how to transform the constraints in Φ into f, respectively f, ψ before applying an EA 2. how to enforce the constraints in S, f, Φ while running an EA Case 1: constraint handling only in the 1st sense (pure penalty approach) Case 2: constraint handling in both senses In Case 2 the question How to solve CSPs by EAs transforms to How to solve COPs by EAs

18 Indirect constraint handling (1) Note: always needed for all constraints in Case 1 for some constraints in Case 2 Some general options a. penalty for violated constraints b. penalty for wrongly instantiated variables c. estimating distance/cost to feasible solution (subsumes a and b)

19 Indirect constraint handling (2) Notation: c i constraints, i = {1,, m} v j variables, j = {1,, n} C j is the set of constraints involving variable v j S c = {z D j1 D jk c(z) = true} is the projection of c, if v j1,, v jk are the var's of c d(s, S c ) := min{d(s, z) z S c } is the distance of s S from S c

20 Indirect constraint handling (3) Formally: a. f ( s) = χ(s,c i m i= 1 w i 1 ) = 0 χ(s,c if s i ), where violates c otherwise i b. f ( s) = n j= 1 w j χ(s,c j ), where χ(s,c j ) = 1 0 if s violates at otherwise least one c C j c. f ( s) = m i= 1 w i d(s,s c i ), where Observe that for each option: s S : Φ( s) f ( s) = 0

21 Indirect constraint handling: Graph coloring example a. b. c. f ( s) = f ( s) = f ( s) = m i= 1 n j= 1 m i= 1 w w i w i j χ(s,c χ(s,c d(s,s i c i ), counts the ' wrong' edges j ), counts the ' wrong' nodes ), counts the ' wrong' edges if we take the number of necessary corrections (recolorings) as distance.

22 Indirect constraint handling: pro s & con s PRO s of indirect constraint handling: conceptually simple, transparent problem independent (at least, problem independent realizations) reduces problem to simple optimization allows user to tune on his/her preferences by weights allows EA to tune fitness function by modifying weights during the search CON s of indirect constraint handling: loss of info by packing everything in a single number said not to work well for sparse problems

23 Direct constraint handling (1) Notes: only needed in Case 2, for some constraints feasible here means Ψ in S, f, Ψ, not Φ in S,, Φ Options: eliminating infeasible candidates (very inefficient, hardly practicable) repairing infeasible candidates preserving feasibility by special operators (requires feasible initial population, (NP-hard for TSP with time windows) decoding, i.e. transforming the search space (allows usual representation and operators)

24 Direct constraint handling (2) PRO's of direct constraint handling: it works well (except eliminating) CON's of direct constraint handling: problem specific no guidelines

25 Evolutionary Computing Evolutionary Art Bu-Ali Sina University Computer Engineering Dep. Fall 2009

26 What is Evolutionary Art? Imagery produced by a process of simulated evolution inside a computer, guided by an artist's aesthetic fitness selection Steven Rooke at allows the artists to generate complex computer artwork without them needing to delve into the actual programming used Andrew Rowbottom at more akin to genetic engineering than to painting Jeffrey Ventrella at

27 What is Evolutionary Art? Technically, it is creating pieces of art through human-computer interaction, where computer: runs evolutionary algorithm human: applies subjective/aesthetic selection

28 The Roles in Evolutionary Art Role of computer: offers choices, creates diversity Role of human: makes choices, reduces diversity Selection (aesthetic, subjective) steers generation process towards implicit user preferences Q: who is creative here?

29 Example: Mondriaan evolver Application evolving images in the style of Piet Mondriaan Programming assignment of my univ. course on evolutionary computing 1999 Dutch-Belgium AI Conference paper On-line toy at: or

30 Mondriaan evolver GUI shows population of 9 pictures User gives grades (thus defines fitness values) Computer performs one evolutionary cycle, i.e. selection, based on this fitness (thus creates mating pool) crossover & mutation (thus creates new population) Repeat

31 The Evolutionary Art Cycle 1 Population Parent pool Parent selection aesthetic selection subjective selection Recombination, mutation

32 Representation in Evolutionary Art Phenotype level User selection acts on this level Genotype level Decoding AGCTCTTA Genetic operators act on this level

33 Mondriaan representation root root root split_y split_y split_y white 0.5 green red 0.33 split_x red 0.33 split_x white 0.5 green white 0.5 split_y yellow 0.5 green

34 The Evolutionary Art Cycle 2 Population phenotypes Parent selection Parent pool phenotypes Decoding Encoding Population genotypes AGCTCTTA CCTTTGGGCCTCACAA TGATCGTA CCTTTGAA AGAGACTA AGAGACTA AGTACTTA GTGACTCC Recomb. mutation AGCTCTTA TGATCGTA GTGACTCC Parent pool genotypes

35 Points of attention Representation phenotypes should be appealing ( fine art ) genotypes should be easy to manipulate (operators) Coding-decoding: should be fast Lamarckian evolution in case of user-defined effects Operators too disruptive: user sees no link between generations too smooth (small changes): evolution is too slow Selection user grades are continuous (fitness values): hard to grade user grades are binary (die/multiply): not enough differentiation

36 Karl Sims, Galápagos Galápagos is an interactive media installation that allows visitors to "evolve" 3D animated forms Exhibited at the: ICC in Tokyo from 1997 to 2000, Interactive Computer Art, Lincoln, Mass. Boston Cyberarts Festival 1999

37 Karl Sims, Galápagos Box insect Beaded arms Multipus-green Jellyfish Bfly larva Multipus-purple

38 Kleiweg, Evolutionary Art in PostScript

39 Eiben et al., Escher evolver Flatfish Exhibited for 6 months in City Museum The Hague Flat screens on walls show computer genarted pictures Visitors vote on separate images (define fitness values) Computer performs one evolutionary cycle every 30 minutes Re-design: visitors choose between two images (split screen)

40 How is this creativity achieved? When evolution is told to build solutions from components, it becomes creative. Only those approaches that use componentbased representations provide sufficient freedom. Evolution now explores new ways of putting components together to construct innovative solutions.

41 Component-based representations Instead of optimising selected elements of a given solution, we allow evolution to build new solutions from scratch, using component-based representations

42 Component-based representations P. Bentley used primitive shapes to construct novel designs

43 Component-based representations John Gero used wall fragments to generate house floor plans

44 Some useful Web links Andrew Rowbottom, Organic, Genetic, and Evolutionary Art (incl. large software overview) Craig Reynolds, Evolutionary Computation and its application to art and design Matthew Lewis, Visual Aesthetic Evolutionary Design Links Steven Rooke, Evolutionary Art, Glossary of Terms: Karl Sims, Homepage at GenArts, Inc., Linda Moss, Evolutionary Graphics