10/20/19 Multi-objective Evolutionary Algorithms Genetic Algorithms - PDF document

10/20/19 Multi-objective Evolutionary Algorithms Genetic Algorithms • Multi-objective optimization problems (MOPs) - Examples - Domination - Pareto optimality - Practical example Multi-objective • EC approaches Optimization - Preference-based - Ideal • Preserving diversity Multi-objective problems Multi-Objective Problems (MOPs) Example: Path planning • Wide range of problems can be categorised by the destination presence of a number of n possibly conflicting objectives: – robotic path planning: – buying a car: speed vs. price vs. reliability – engineering design: lightness vs. strength source • Solving an MOP presents two problems: – finding set of good solutions Goal: find a shortest, obstacle-avoiding path from – choice of best for particular application source to destination. 1

10/20/19 Multi-objective problems Multi-objective problems Example: Path planning Example: Path planning Conflicting objectives: • What are the objectives? – Path length (minimize) – Obstacle collisions (minimize) destination • Any others? – Number of waypoints (minimize) – Smoothness (minimize/maximize – depends on source definition) – Intermediate destinations? Optimal for obstacle collisions Optimal for path length Which is a better solution? Multi-objective problems Multi-objective Optimization Problems Example: Buying a car Two spaces Decision (variable) Objective space space Fast but expensive speed Inexpensive cost but slow Which is a better solution? 2

10/20/19 Multi-objective Optimization Problems Multi-objective Optimization Problems Comparing Solutions The Dominance relation • Solution X dominates solution Y, (X Y), if: • Optimisation task: Objective space – X is no worse than Y in every objective Minimize both f 1 and f 2 ∀" ∈ 1, …, ' ( ) ≤ + ) , and ∃" ∈ 1, …, ' ( ) < + ) – X is better than Y in at least one objective • Then: a is better than b solutions a is better than c dominated by x a is worse than e a and d are incomparable Important note: above definition is solutions for minimization problem. Reverse dominating inequalities for maximization x Multi-objective Optimization Problems Multi-objective Optimization Problems Origins of Pareto optimization Pareto optimality • Vilfredo Pareto (1848-1923) was an Italian economist, • Solution x is non-dominated among a set of solutions Q political scientist and philosopher if no solution from Q dominates x • For much of his life he was a political economist at the University of Lausanne (Switzerland) • A set of non-dominated solutions from the entire • Manual of Political Economy (1906): described feasible solution space is the Pareto-optimal set, equilibrium for problems consisting of a system of its members Pareto-optimal solutions objectives and constraints • Pareto optimality(economics): an economy is is • Pareto-optimal front: an image of the Pareto-optimal set functioning optimally when no one’s position can be in the objective space improved without someone else’s position being made worse 3

10/20/19 Multi-objective Optimization Problems Multi-objective Optimization Problems Illustration of the concepts Illustration of the concepts f 2 (x) f 2 (x) min min non-dominated solutions f 1 (x) f 1 (x) min min Practical Example: Practical Example: The beam design problem The beam design problem – Formal Definition p d 2 = r • Minimize f d l ( , ) l (beam weight) 1 4 Minimize weight and deflection of a beam (Deb, 2001): 3 64 Pl • minimize = d = f ( , ) d l (beam deflection) 2 p 4 3 E d • subject to £ £ 0.01 m d 0.05 m £ £ 0.2 m l 1.0 m d 32 Pl (maximum stress) s = £ S max p 3 y d d £ d max where r = 3 = 7800 kg/m , P 2 kN = E 207 GPa = d = S 300 MPa, 0.005 m y max 4

10/20/19 Practical Example: Practical Example: The beam design problem The beam design problem Feasible Solutions: Goal: Finding non-dominated solutions: Decision (variable) space Objective space Goal of multi-objective optimizers Single vs. Multi-objective Optimization • Find a set of non-dominated solutions (approximation of Characteristic Singleobjective Multiobjective the Pareto-optimal front) following the criteria of: optimisation optimisation – convergence (as close as possible to the Pareto- optimal front) Number of objectives one more than one – diversity (spread, distribution) Spaces single two: decision (variable) space, objective space Comparison of x is better than y x dominates y candidate solutions Result one (or several equally Pareto-optimal set good) solution(s) Algorithm goals convergence convergence, diversity 5

10/20/19 Multi-objective optimization Multi-objective optimization Two approaches Preference-based approach • Given a multiobjective optimisation problem, • Preference-based: traditional, using single objective optimisation methods • use higher-level information on importance of objectives • to transform the problem into a singleobjective one, • Ideal: possible with novel multiobjective optimisation techniques, • then solve it with a single objective optimization method enabling better insight into the problem • to obtain a particular trade-off solution. Multi-objective optimization Multi-objective optimization Preference-based approach Ideal approach • Given a multiobjective optimization problem, • solve it with a multi-objective optimization method Hyperplanes in the objective space! • to find multiple trade-off solutions, • and then use higher-level information M M å å x x = Î = Modified problem: F ( ) w f ( ), w [0,1], w 1 m m m m • to obtain a particular trade-off solution. m = 1 m = 1 The weighted sum scalarizes the objective vector: we no have a single-objective problem 6

10/20/19 EC approach to multi-objective optimization: EC approach to multi-objective optimization: Advantages Requirements • Population-based nature of search means you can • Way of assigning fitness, simultaneously search for set of points approximating – usually based on dominance Pareto front • Preservation of diverse set of points • Can return a set of trade-off solutions (approximation – similarities to multi-modal problems set) in a single run • Don’t have to make guesses about which combinations • Remembering all the non-dominated points you have of weights might be useful seen • Makes no assumptions about shape of Pareto front - can – usually using elitism or an archive be convex / discontinuous etc. EC approach: EC approach: Fitness assignment options Diversity maintenance • Could use aggregating approach and change weights • Usually done by niching techniques such as: during evolution – fitness sharing – no guarantees – adding amount to fitness based on inverse distance to nearest neighbour (minimisation) – (adaptively) dividing search space into boxes and counting • Different parts of population use different criteria occupancy – e.g. VEGA, but no guarantee of diversity • All rely on some distance metric in genotype / phenotype • Dominance space – ranking or depth based – fitness related to whole population – Question: how to rank non-comparable solutions? 7

10/20/19 EC approach: Multi-objective optimization Problem Remembering good solutions Summary • Could just use elitist algorithm • MO problems occur very frequently – e.g. ( µ + l ) replacement • EAs are very good at solving MO problems • Maintain an archive of non-dominated solutions – some algorithms use this as second population that can be in recombination etc. • MOEAs are one of the most successful EC subareas – others divide archive into regions too, e.g. PAES 8