An Overview of Evolutionary Multi-Objective Optimization...Mathematical Programming Techniques...

An Overview of EvolutionaryMulti-Objective Optimization

Carlos A. Coello Coello

[email protected]

Evolutionary Computation Group (EVOCINV)Computer Science Department

Av. IPN No. 2508, Col. San Pedro ZacatencoMexico, D.F. 07360, MEXICO

Metaheuristic Optimization, Machine Learning, and AIMarch 2021

Carlos A. Coello Coello An Overview of Evolutionary Multi-Objective Optimization

Motivation

Most problems in nature have several (possibly conflicting) objectives to besatisfied (e.g., design a bridge for which want to minimize its weight and costwhile maximizing its safety). Many of these problems are frequently treatedas single-objective optimization problems by transforming all but oneobjective into constraints.


Formal Definition

Find the vector ~x∗ =[x∗

1 , x∗2 , . . . , x

∗n]T which will satisfy the m

inequality constraints:

gi(~x) ≤ 0 i = 1,2, . . . ,m (1)

the p equality constraints

hi(~x) = 0 i = 1,2, . . . ,p (2)

and will optimize the vector function

~f (~x) = [f1(~x), f2(~x), . . . , fk (~x)]T (3)


Notion of Optimality in MOPs

Having several objective functions, the notion of “optimum” changes, becausein MOPs, the aim is to find good compromises (or “trade-offs”) rather than asingle solution as in global optimization.

The notion of “optimum” that is most commonly adopted is that originallyproposed by Francis Ysidro Edgeworth (in 1881) in his book entitledMathematical Psychics.



This notion was generalized by the italian economist Vilfredo Pareto (in 1896)in his book Cours d’Economie Politique. Although some authors callEdgeworth-Pareto optimum to this notion (originally called ophelimity) it isnormally preferred to use the most commonly accepted term: Paretooptimum.



Pareto OptimalityWe say that a vector of decision variables ~x∗ ∈ F is Pareto optimal if theredoes not exist another ~x ∈ F such that fi(~x) ≤ fi(~x∗) for all i = 1, . . . , k andfj(~x) < fj(~x∗) for at least one j (assuming that all the objectives are beingminimized).

In words, this definition says that ~x∗ is Pareto optimal if there exists nofeasible vector of decision variables ~x ∈ F which would decrease somecriterion without causing a simultaneous increase in at least one othercriterion. This concept normally produces a set of solutions called the Paretooptimal set. The vectors ~x∗ corresponding to the solutions included in thePareto optimal set are called nondominated. The image of the Paretooptimal set is called the Pareto front.


Mathematical Programming Techniques

Currently, there are over 30 mathematical programmingtechniques for multiobjective optimization. However, thesetechniques tend to generate elements of the Pareto optimal setone at a time. Additionally, most of them are very sensitive tothe shape of the Pareto front (e.g., they do not work when thePareto front is concave or when the front is disconnected).


Evolutionary Algorithms

The idea of using techniques based on the emulation of the mechanism ofnatural selection to solve problems can be traced as long back as the 1930s.However, it was not until the 1960s that the three main techniques based onthis notion were developed: genetic algorithms [Holland, 1962], evolutionstrategies [Schwefel, 1965] and evolutionary programming [Fogel, 1966].These approaches are now collectively denominated evolutionaryalgorithms.



Evolutionary algorithms operate on a set of solutions (called population).Initially, the elements of this population (called individuals) are randomlygenerated. Each member of the population encodes all the decision variablesof the problem to be solved. In order to assess how good a solution is, weneed to define a fitness function. Individuals with the highest fitness have ahigher probability of being selected to become parents so thatrecombination can be applied. The offspring produced are then mutatedand the procedure is repeated several times (according to a parameter thatdefines the maximum number of generations).



Evolutionary algorithms seem particularly suitable to solvemultiobjective optimization problems, because of theirpopulation-based nature. This allows us to find severalmembers of the Pareto optimal set in a single run of thealgorithm, instead of having to perform a series of separateruns as in the case of the traditional mathematical programmingtechniques. Additionally, evolutionary algorithms are lesssusceptible to the shape or continuity of the Pareto front (e.g.,they can easily deal with discontinuous or concave Paretofronts), whereas these two issues are a real concern formathematical programming techniques.


A Taxonomy of MOEAs

The Old Days (1980s to early 1990s)Non-Elitist Non-Pareto-based Methods

Lexicographic OrderingLinear Aggregating FunctionsVEGAε-Constraint MethodTarget Vector Approaches


A Taxonomy of MOEAs

The Old Days (early 1990s to mid 1990s)Non-Elitist Pareto-based Methods

Pure Pareto rankingMOGANSGANPGA and NPGA 2


A Taxonomy of MOEAs

Contemporary Approaches (late 1990s to early 2000s)Elitist Pareto-based Methods

SPEA and SPEA2NSGA-IIPAES, PESA and PESA IIMicro-genetic Algorithm for Multi-Objective Optimizationand µGA2

Many others (most of them already forgotten...)


A Taxonomy of MOEAs

Recent Approaches (from early 2000s until today)MOEA/D (and its many variants)Indicator-Based Approaches

SMS-EMOAHyPEOther Approaches

NSGA-III (and its many variants)


A Taxonomy of MOEAs

In general, modern MOEAs consist of two basic components:

A selection mechanism that normally (but not necessarily) incorporatesPareto optimality.

A density estimator, which is responsible for maintaining diversity, andtherefore, keeping the MOEA from converging to a single solution.


Selection Mechanisms

There are three main types of MOEAs in current use:Pareto-based MOEAsDecomposition-based MOEAsIndicator-based MOEAs



Pareto-based MOEAsThese are the traditional MOEAs in which the selectionmechanism is based on Pareto optimality. Most of them adoptsome form of nondominated sorting and a density estimator(e.g., crowding, fitness sharing, entropy, adaptive grids, parallelcoordinates, etc.).

Main limitationsScalability in objective function space is clearly a limitation ofPareto-based MOEAs unless a significantly larger populationsize is adopted. Another alternative is to change the densityestimator, but most people don’t seem to be interested inmoving in that direction.



Decomposition-based MOEAsThe core idea of these approaches is to transform amulti-objective problem into several single-objectiveoptimization problems which are simultaneously solved usinginformation from its neighboring subproblems.

Main limitationsThe performance of decomposition-based MOEAs relies on thescalarizing function that they adopt. They are also sensitive tothe method used to generate weights. However, they arescalable in objective function space (although an increase inthe number of objectives will increase the population size).



Indicator-based MOEAsThe original idea was to adopt a performance indicator for theselection mechanism of a MOEA. However, some researchersdiscovered that the mere use of a performance indicator in thedensity estimator was enough to have a good performance(e.g., SMS-EMOA).

Main limitationsThe only performance indicator which is known to be fullyPareto compliant is computationally expensive in highdimensionality (in objective space). Other performanceindicators are available, some of which are weakly Paretocompliant (e.g., R2 and IGD+). However, researchers don’tseem to like them much.


Density Estimators

This is an important component of modern MOEAs which isrequired to generate several elements of the Pareto optimal setin a single run. The main methods that have been adopted arethe following:

Fitness sharingClusteringEntropyAdaptive gridsCrowdingPerformance indicatorsParallel coordinates


Number of papers published per year (late 2020)


Some Sample Applications

Van der Blom et al. [2019] developed an approach that was able to learnheuristic rules on building spatial design by data-mining multi-objectiveoptimization results. Then, this information was used to gain new insights thatcan help architects to build better spatial designs with respect to thermal andstructural performance.

Koen van der Blom, Sjonnie Boonstra, Herm Hofmeyer and Michael Emmerich, “Analysing Optimisation Data forMulticriteria Building Spatial Design”, in Kalyanmoy Deb et al. (Editors), Evolutionary Multi-CriterionOptimization, 10th International Conference, EMO 2019, pp. 671–682, Springer. Lecture Notes in ComputerScience Vol. 11411, East Lansing, Michigan, USA, March 10–13, 2019.



Cabrera et al. [2018] adopted a local search procedure and a multi-objectivemetaheuristic to optimize the intensity-modulated radiation therapy (IMRT) forcancer treatment. When treating this as a multi-objective problem, the idea isto lead to a set of dose distributions that, depending on both doseprescription and physician preferences, can be selected as the preferredtreatment for a patient.

Guillermo Cabrera, Matthias Ehrgott, Andrew J. Mason and Andrea Raith, “A Metaheuristic Approach to Solvethe Multiobjective Beam Angle Optimization Problem in Intensity-Modulated Radiation Therapy”,International Transactions in Operational Research, Vol. 25, No. 1, pp. 243–268, January 2018.



Xiao et al. [2018] proposed an extension of MOEA/D which is based onreference distance to solve software project portfolio optimization problems.In this problem, the aim is that a large software company selects their projectportfolios to gain maximum return with limited resources under manyconstraints.

Jing Xiao, Jing-Jing Li, Xi-Xi Hong, Min-Mei Huang, Xiao-Min Hu, Yong Tang and Chang-Qin Huang, “An ImprovedMOEA/D Based on Reference Distance for Software Project Portfolio Optimization”, Complexity, ArticleNumber: 3051854, 2018.



Khayyam et al. [2020] provide a review of concepts on machine learningtechniques that can deal with modeling big or limited data and presents acase study (using real data collected from a Carbon Fiber production line)related to the use of a hybrid algorithm for Industry 4.0 applications withlimited data. The authors present an intelligent algorithm for robust datamodeling of nonlinear systems based on input-output data. The algorithmincorporates a multi-objective genetic algorithm for designing the topology ofthe neural network adopted by the authors and to overcome overfitting.

Hamid Khayyam, Ali Jamali, Alireza Bab-Hadiashar, Thomas Esch, Seeram Ramakrishna, Mahdi Jalili and MinooNaebe, “A Novel Hybrid Machine Learning Algorithm for Limited and Big Data Modeling With Application inIndustry 4.0”, IEEE Access, Vol. 8, pp. 111381–111393, 2020.



Goti et al. [2019] use a multi-objective genetic algorithm to simultaneouslyoptimize costs and benefits of condition-based maintenance. The goal is todetermine the optimal values of preventive intervention limits for equipmentunder corrective and preventive maintenance cost criteria. As a case study,the authors consider an optimization problem that consists of threecontinuously monitored components of a simplified injection system.

Aitor Goti, Aitor Oyarbide-Zubillaga, Elisabete Alberdi, Ana Sanchez and Pablo Garcia-Bringas, “OptimalMaintenance Thresholds to Perform Preventive Actions by Using Multi-Objective Evolutionary Algorithms”,Applied Sciences-Basel, Vol. 9, No. 15, Article number: 3068, August 1, 2019.


Research representative of the current trends

Schemes for solving many-objective problems (i.e., with 4or more objectives) efficiently and effectively.Hybridizations of MOEAs with mathematical programmingtechniques (e.g., gradient-based methods).Use of surrogate and parallel methods to deal withexpensive objective functions.Novel applications (e.g., in bioinformatics, medicine,operating systems, etc.).Solution of multi-modal multi-objective optimizationproblems.


Future Challenges

Can we combine different operators and algorithmiccomponents to produce tailored (and highly competitive)multi-objective evolutionary algorithms?Can we design selection mechanisms for multi-objectiveevolutionary algorithms different from the ones currentlyavailable?Can we design better density estimators and performanceindicators for many-objective optimization problems?Specialized schemes (e.g., based on cooperativecoevolution) for dealing with large scale MOEAs (e.g., with5,000 or 10,000 decision variables)Multi-objective hyper-heuristics for continuous optimization.


To know more about evolutionarymulti-objective optimization

Please visit the new webpage of the EMOO repository:

https://emoo.cs.cinvestav.mx/


To know more about evolutionarymulti-objective optimization

The EMOO repository currently contains:

Over 12,570 bibliographic references including 303 PhDtheses, 53 Masters theses, over 6155 journal papers andover 4495 conference papers.Contact information of 79 EMOO researchers.Public domain implementations of SPEA, NSGA, NSGA-II,the microGA, MOGA, ε-MOEA, MOPSO and PAES, amongothers.


An Overview of Evolutionary Multi-Objective Optimization...Mathematical Programming Techniques...

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