Multi-objective Evolutionary Algorithms (for NACST/Seq) summarized by Shin, Soo-Yong

© 2002, SNU BioIntelligence Lab, Reference Multi-Objective Optimization using Evolutionary Algorithms, Kalyanmoy Deb, John Wiley & Sons, LTD., 2002

© 2002, SNU BioIntelligence Lab, Multi-Objective Optimization Optimization problems with multiple, conflicting objectives. M objective functions:

© 2002, SNU BioIntelligence Lab, Decision Space vs. Objective Space

© 2002, SNU BioIntelligence Lab, Objectives in Multi-Objective Optimization Two goals in a MOO  To find a set as close as possible to the Pareto- optimal front  To find a set of solutions as diverse as possible  Ex) airline route: cost vs. time

© 2002, SNU BioIntelligence Lab, Difference with Single-Objective Optimization Two goals instead of one Dealing with two search space  objective space & decision space (for SOO) No artificial fix-ups  cf) weight-sum, ε-constraints method..

© 2002, SNU BioIntelligence Lab, Concept of Domination A solution x (1) is said to dominate the other solution x (2), if both conditions 1 and 2 are true: 1.The solution x (1) is no worse that x (2) in all objectives 2.The solution x (1) is strictly better than x (2) in at least one objective

© 2002, SNU BioIntelligence Lab, Dominance Example

© 2002, SNU BioIntelligence Lab, Pareto-Optimality Non-dominated set  Among a set of solutions P, the non-dominated set of solutions P’ are those that are not dominated by any member of the set P. Globally Pareto-optimal set  The non-dominated set of the entire feasible search space S is the globally Pareto-optimal set  First level non-dominate front. Locally Pareto-optimal set

© 2002, SNU BioIntelligence Lab, Pareto-Optimality Example

© 2002, SNU BioIntelligence Lab, Non-dominated Sorting of a Population 1. Set all non-dominated sets P j, (j=1,2,…) as empty sets. Set non-domination level counter j = Find the non-dominated set P’ of P 1.Set solution counter i=1 and create an empty non-dominate set P’. 2.For a solution j ∈ P (but j ≠ i), check if solution j dominates solution I, If y4s, go to Step If more solutions are left in P, increment j by one and go to Step 2-2; otherwise, set P’=P’ ∪ {i}. 4.Increment I by one. If i ≤N, go to Step 2-2; otherwise stop and declare P’ as the non-dominated set. 3. Update P j = P’ and P = P\P’. 4. If P ≠ Φ, increase j by one and go to Step 2. Otherwise, stop and declare all non-dominated sets P i, for i=1,2,…,j.

© 2002, SNU BioIntelligence Lab, Non-dominates Sorting Example

© 2002, SNU BioIntelligence Lab, Classical Methods for MOO Weighted sum method ε-constraints method Weighted metric methods Benson’s method Value function method Goal programming methods Interactive methods

© 2002, SNU BioIntelligence Lab, Evolutionary Algorithms Multi-modal function optimization  Multi-modal functions have multiple optimum solutions, of which many are local optimal solutions  Diversity through mutation  Preselection  Crowding model  Sharing function model Crowding & sharing function model are useful to MOEA

© 2002, SNU BioIntelligence Lab, Non-Elitist Multi-Objective EA Motivations:  A user is usually not sure of an exact trade-off relationship among objectives.  Equi-spaced weight does not always result to equi-spaced trade-off solutions.  Especially, in non-linear problems. After finding diverse set of optimal solutions, it is possible to calculate the associated weights.  Enables to choose from different trade-offs.

© 2002, SNU BioIntelligence Lab, MOEA: Early Suggestions VEGA (Schaffer, 1984)  First real implementation of a multi-objective evolutionary algorithm.  Bias towards independent champions. Goldberg, 1989  Suggested the concept of domination.  Use of niching strategy among solutions of a non- dominated class.  Had impact on MOGA, NPGA, NSGA

© 2002, SNU BioIntelligence Lab, Vector Evaluated Genetic Algorithm Evaluated an objective vector instead of a scalar objective function.  Each element of the vector represents each objective function. Simplest, straightforward extension of GA.

© 2002, SNU BioIntelligence Lab, Vector Optimized Evolution Strategy Modification to basic self-adaptive evolution strategy for single-objective. Solution is represented by using a diploid chromosome (dominant, recessive string). Keeps external set of non-dominated solutions.  Does not take part in genetic operations. No further study.

© 2002, SNU BioIntelligence Lab, Weight-Based Genetic Algorithm GA string represents both decision variable and weights.  Fitness: weighted sum of objectives. Maintain diversity in the weight vectors among population.  Niching method on substring for weights – sharing function approach  Subpopulation for different pre-defined weight vectors – vector evaluated approach

© 2002, SNU BioIntelligence Lab, Random Weighted GA Similar to WBGA Each solution is associated with random normalized weight vector.  Emphasize solutions which may lead to different solutions in Pareto-optimal region. Unable to find Pareto-optimal solutions in non- convex problems.

© 2002, SNU BioIntelligence Lab, Multiple Objective GA Use the non-dominated classification of a GA population. Explicitly caters to emphasize non-dominated solutions and simultaneously maintains diversity in the non-dominated solutions.

© 2002, SNU BioIntelligence Lab, Non-Dominated Sorting GA Before selection, the population is ranked on the basis of domination (Pareto ranking) All nondominated individuals are classified into one category. To maintain the diversity of the population, these classified individuals are shared with their dummy fitness values

© 2002, SNU BioIntelligence Lab, Niched-Pareto GA Uses a binary tournament selection scheme based on Pareto dominance. Solutions are selected if they dominate both the other and some small group of randomly selected solutions, but fitness sharing occurs only in the cases when both solutions are (non)dominated.

© 2002, SNU BioIntelligence Lab, Predator-Prey ES Concept of predator-prey model is used. This algorithm does not use a domination check to assign fitness to a solution.

© 2002, SNU BioIntelligence Lab, Elitist MOEA The presence of elites  GAs converge to the global optimal solution  enhance the probability of creating better offspring Which solutions are elites in the context of multi-objective optimization?  A solution can be evaluated based on non- domination rank in the population

© 2002, SNU BioIntelligence Lab, Elitist NSGA: NSGA-II Uses an explicit diversity-preserving mechanism.

© 2002, SNU BioIntelligence Lab, Distance-Based Pareto GA Progress towards the Pareto-optimal front Maintain diversity among solutions Elite size is not restricted  Increase the complexity Fitness assignment scheme is sensitive to the ordering of individuals in a population

© 2002, SNU BioIntelligence Lab, Strength Pareto EA Uses an archive containing non-dominated solutions previously found. At each generation, non-dominated individuals are copied to the external non- dominated set. For each individual, a strength value is computed (ranking value). Clustering technique is used to keep diversity.

© 2002, SNU BioIntelligence Lab, Thermodynamical GA The fitness function is motivated from the thermodynamic equilibrium condition

© 2002, SNU BioIntelligence Lab, Pareto-Archived ES (1+1)-ES Uses only mutation Single parent and a single offspring

© 2002, SNU BioIntelligence Lab, Multi-Objective Messy GA MOO version of Messy GA

© 2002, SNU BioIntelligence Lab, Constrained MOEA Constraints  Divides search spaces into two divisions  Feasible vs. infeasible regions  Equality vs. inequality  Hard vs. soft

© 2002, SNU BioIntelligence Lab, Constraint Handling MOEAs Ignoring infeasible solutions  Difficult to find any feasible solution. Penalty function approach Jiménez-Verdegay-Goméz-Skarmeta’s method Constrained tournament approach Ray-Tai-Seow’s method

© 2002, SNU BioIntelligence Lab, Salient Issues of MOEA Illustrative representation of non-dominated solutions Development of performance measures Test problem design for unconstrained and constrained multi-objective optimization Comparative studies of different MOEAs Decision variable vs. objective space niching Preference of a particular region in the Pareto- optimal front Single-objective constraint handling using MOEAs Scaling issues of MOEAs in more than two objectives Design of convergent MOEAs Controlled elitism in elitist MOEAs Design of MOEAs for scheduling problems.

© 2002, SNU BioIntelligence Lab, Difficulties in Converging to the Pareto-Optimal Front Multi-modality Deception Isolated optimum Collateral noise

© 2002, SNU BioIntelligence Lab, Searching for Preferred Solutions How does one choose a particular solution from the obtained set of non-dominated solutions? Post-optimal techniques  Compromise programming  Pseudo-weight vector approach Optimization-level techniques  Utility functions  Biased sharing approach  Guided domination approach  Weighted domination approach

© 2002, SNU BioIntelligence Lab, Scaling Issues The problem difficulties varies rather interestingly with the number of objectives.

© 2002, SNU BioIntelligence Lab, Scaling: Non-Dominated Solutions in a Population As increase the number of objectives, the number of non-dominated solutions in the initial random population will also increase.

© 2002, SNU BioIntelligence Lab, Scaling: Population Sizing Too many initial non-dominated solution..  Increase the population size  Modify algorithm  About 30% of initial pop is good. Objective space sharing vs. Parameter space sharing Tricky generation of initial population (if the information is given) Dynamic population sizing

© 2002, SNU BioIntelligence Lab, Controlling Elitism To ensure better convergence, a search algorithm may need diversity in both aspects – along the Pareto-optimal front and lateral to the Pareto- optimal front.

© 2002, SNU BioIntelligence Lab, Controlling Elitism in NSGA-II Restrict the number of individuals in the current best non-dominated front adaptively. Maintain a predefined distribution of number of individuals in each front.

© 2002, SNU BioIntelligence Lab, Controlling Elitism in NSGA-II 1. The population R t =P t  Q t 1.pop size = 2N, number of front = k 2.If there are more solutions than allowed, choose N i solutions by using the crowded tournament selection. 3.Otherwise, choose all solutions and count the number of remaining slots. The maximum allowed number of individuals in the next front is increased by remaining slots.

Non-dominated Sorting Genetic Algorithm (NSGA)

© 2002, SNU BioIntelligence Lab, NSGA procedure 1. Sort the population P according to non-domination  All solutions in the first set belong to the best non- dominated set in the population 2. Fitness assignment 1.Choose sharing parameter σ share and a small positive number ε and initialize F min = N + ε. Set front counter j = 1/ 2.Classify population P according to non-domination: 3.For each q ∈ P j 1. Assign fitness F j (q) = F min - ε. 2. Calculate niche count nc q using equation

© 2002, SNU BioIntelligence Lab, NSGA procedure 3. Calculate shared fitness F j ’ (q) = F j (q) /nc q 4.F min = min(F j ’ (q) : q ∈ P j ) and set j=j+1. 5.If j ≤ ρ, go to Step 3. Otherwise, the process is complete.

© 2002, SNU BioIntelligence Lab, NSGA Advantages  The assignment of fitness according to non- dominated sets  Since better non-dominated sets are emphasized systematically, an NSGA progresses to the Pareto- optimal region.  Performing sharing in the parameter space allows phenotypically diverse solutions to emerge.

© 2002, SNU BioIntelligence Lab, NSGA Disadvantages  The sharing function approach requires the sharing parameter.  Performance of an NSGA is sensitive to the sharing parameter

© 2002, SNU BioIntelligence Lab, NSGA-II: Elitist NSGA

© 2002, SNU BioIntelligence Lab, NSGA-II Procedure P t : parent population Q t : offspring population R t : P t  Q t 1. Create R t, perform a non-dominated sorting to R t, identify different fronts F i 2. P t +1 = , i=0. Until | P t +1 | + | F i | < N, perform P t +1 = P t +1  F i i=i+1

© 2002, SNU BioIntelligence Lab, NSGA-II Procedure 3. Perform the crowding-sort (F i, <c), include the most widely spread (N-|P t +1|) solutions 4. Create Q t +1 from P t +1 (using the crowed tournament selection, crossover, mutation)

© 2002, SNU BioIntelligence Lab, NSGA-II Procedure: Crowded Tournament Selection Operator A solution i wins a tournament with another solution j if any of the following conditions are true :  If solution i has a better rank, that is, r i < r j  If they have the same rank but solution i has a better crowding distance than solution j, that is, r i = r j and d i > d j

© 2002, SNU BioIntelligence Lab, NSGA-II Procedure: Crowded Tournament Selection Operator 1. Call the number of solutions in F as l = |F|. For each i in the set, first assign d i =0. 2. For each objective function sort the set in worse order, or find the sorted indices vector: I m = sort (f m, >). 3. Assign a large distance to the boundary solutions, and for all other solutions j=2 to (l-1) :

© 2002, SNU BioIntelligence Lab, NSGA-II Advantages  No extra niching parameter is required  Crowding distance can be implemented in the parameter space Disadvantages  If pop size is small, NSGA-II shows the poor exploration power.

© 2002, SNU BioIntelligence Lab, Controlled Elitist NSGA-II Restrict the number of individuals in the current best non-dominated front adaptively. Maintain a predefined distribution of number of individuals in each front.

© 2002, SNU BioIntelligence Lab, Controlled Elitist NSGA-II

© 2002, SNU BioIntelligence Lab, Controlled Elitist NSGA-II 1. The population R t =P t  Q t 1.pop size = 2N, number of front = k 2.If there are more solutions than allowed, choose N i solutions by using the crowded tournament selection. 3.Otherwise, choose all solutions and count the number of remaining slots. The maximum allowed number of individuals in the next front is increased by remaining slots.

© 2002, SNU BioIntelligence Lab, NACST/Seq Controlled elitist NSGA-II with clustering  Simple k-means clustering with threshold