Research Issues in Many-Objective Optimization with Evolutionary Algorithms Frederico Gadelha Guimarães fredericoguimaraes@ufmg.br +55 31-3409-3419 Faculty of Engineering Department of Electrical Engineering Universidade Federal de Minas Gerais
Presentation plan • Introduction and terminology • Motivation • Issues in many-objective optimization • Approaches and techniques • Directions
Introduction and terminology Multi-objective optimization problems:
Introduction and terminology Multi-objective optimization problems: Objective 2 worse Objective 1 better worse
Introduction and terminology • From a multi-objective problem to a single objective problem: preferences and aggregation methods; • The optimization process returns a single solution;
Introduction and terminology • Since evolutionary algorithms work with a population of points, they can search for a representative set of estimates of Pareto optimal solutions; • Multi-objective Evolutionary Algorithms (MOEAs): ← • P(t+1) Sv{ V{ Sr{P(t)} }, P(t) }
Advantages of searching for the trade-off front • Preferences do not need to be specified a priori – choose after seeing the alternatives; • Offer different alternatives to different clients; • Reveal common properties among trade-off solutions; • Introduce more flexibility into the design process;
A brief history • 1984: first EMO approaches; • 1990: dominance-based ranking; • 1990: dominance-based ranking with diversity preservation techniques; • 1995: elitist algorithms; convergence proofs; preference incorporation; • 2000: comparison and performance; test functions; quality measures; • 2000: EMO+MCDM; indicator-based algorithms • 2010: statistical performance evaluation; • 2010: scalability; many-objective optimization;
What about scalability? First of all: how many is too many? It was only in recent years that researchers have investigated the scalability of MOEAs – and the results were not favourable: • Khare et al. (2003): the poor scalability of NSGA-II, PESA and SPEA2 in scalable test functions; • Hughes (2005): aggregation methods with multistart perform better MOEAs; • Knowles & Corne (2007): MOEAs do not perform better than random search in problems with more than 10 objectives; • Purshouse & Fleming (2007): the ability of variation operators to produce solutions that dominate their parents decrease with increasing the number of objectives;
What about scalability? Understanding the problem Garza-Fabre et al. (2011):
What about scalability? Difficulties with many-objective optimization: • Loss of selective pressure (proximity and diversity); • Dimensionality and computational cost; • Visualization of solutions; • Decision-making under a huge set of alternatives;
Why many-objective optimization? • Multiobjectivization: supplementary objectives or decomposition of the original objective; • Constraint-handling; • Multidisciplinary optimization (MDO), e.g. aircraft design;
Why many-objective optimization? • Multiobjectivization: supplementary objectives or decomposition of the original objective; • Constraint-handling; • Multidisciplinary optimization (MDO), e.g. aircraft design; • Musselman & Talavage (1980): Water resource engineering problem with 5 objectives and 7 constraints; • Fleming et al. (2005): a flight control system with 8 objectives; • Hughes (2007): Radar waveform optimization with 9 objectives; • Sulflow et al. (2007): Nurse scheduling problem with 25 objectives; • Knowles & Corne (2007): Travelling salesman problems and job shop scheduling problems with 5 to 20 objectives;
Different notions of dominance • Pareto dominance; • Epsilon dominance; • Cone dominance;
Different notions of dominance • Batista et al. (EMO 2011): Pareto cone epsilon dominance: relaxation of dominance that enables the approximation of nondominated points in some adjacent boxes that would otherwise be epsilon-dominated
Different notions of dominance
Different notions of dominance • Batista et al. (IEEE CEC 2011) • Order induced by different dominance criteria in some quadratic test problems and DTLZ problems; (1)Rate of nondominated solutions (RNS): proportion of points within a given finite set that are not dominated by any other point in that set; (2)Normalized dominance depth (NDD): Number of successive fronts that can be obtained from a given finite set of points, divided by the size of the test set;
Different notions of dominance • Batista et al. (IEEE CEC 2011) • Order induced by different dominance criteria in some quadratic test problems and DTLZ problems;
Approaches – modifying Pareto dominance • Sato et al. (2007): use cone dominance to improve convergence: • Increases selective pressure, but decreases diversity;
Approaches – modifying Pareto dominance • Saxena et al. (2009): uses epsilon dominance to improve convergence together with PCA-based approach for dimensionality reduction; • Argues that epsilon dominance offers a good balance between convergence and diversity;
Approaches – modifying ranking • Drechsler et al. (2001): proposes the relation “favour”, based on the number of objectives for which one solution is better than the other; • Zou et al. (2008): introduce L-dominance: X1 L-dominates X2 if: • B(X1,X2) – W(X1,X2) > 0; • The p-norm of F(X1) is smaller than the p-norm of F(X2);
Approaches – modifying ranking • Sato et al. (2009): introduce Pareto partial dominance: • Select r<m objectives to check Pareto dominance and rank the population; • At every fixed number of generations, switch the r objective functions used for ranking;
Approaches – modifying ranking • Wang & Wu (2007): introduce fuzzy Pareto-dominance: X1 fuzzy-dominates X2 with degree: μ( X 1, X 2 )= 1 N ∑ μ b ( f i ( X 1 )− f i ( X 2 ))+ 1 2N ∑ μ e ( f i ( X 1 )− f i ( X 2 ))
Approaches – modifying ranking • Knowles & Corne (2007): simple average ranking performs better than more complicated ranking schemes; Simple average ranking: each solution is ranked according to each objective, then the average rank is computed for each solution; However, the gain in selective pressure towards proximity to the Pareto front comes at the expense of diversity: few solutions are found;
Approaches – performance indicators • Indicator-based Evolutionary Algorithms (IBEA): Zitzler et al. (2004); • Hypervolume-based MOEAs: Beume et al. (2007); • The search ability of IBEAs scales well with the number of objectives; • However, the time to compute the hypervolume grows exponentially with the number of objectives – impractical for more than six objectives; • Brockhoff & Zitzler (2006, 2007): dimensionality reduction to extend the applicability of hypervolume-based MOEA; • Tagawa et al. (2011): multi-core processing to reduce the cost of hypervolume computation; • Many papers on computing hypervolume...
Approaches – dimensionality reduction Saxena & Deb (2007): dimensionality reduction using PCA methods: • Sources of redundancy of objectives: – either non-conflicting objectives or – the removal of a given objective from the problem makes no significant difference in the front obtained; • Run a MOEA for a large number of generations then reduce the number of objectives using the correlation matrix of the objective values, while maintaining the shape of the Pareto front;
Approaches – dimensionality reduction Brockhoff & Zitzler (2006, 2007): dimensionality reduction based on dominance, however high computational cost; Singh et al. (2011): similar ideas but using heuristics instead: • Relevant or critical objectives are the ones that affect more the number of nondominated solutions in the population; • Run a MOEA for a large number of generations then reduce the number of objectives using the following heuristic: compute the change in the number of nondominated solutions with the removal of a given objective; and eliminate the objective that causes negligible change in the number of nondominated solutions;
Approaches – scalar functions Scalar functions provide a way to aggregate objectives without the computational cost of hypervolume indicators; • Hughes (2007): different scalar functions are defined and each solution is ranked according to each scalar function. An overall rank is calculated based on the multiple ranks; • Ishibuchi et al. (2006, 2007): different scalar functions are defined, but each solution is evaluated with a single scalar function; • Wickramasinghe et al. (2009): distance to reference points to guide PSO in many-objective optimization;
Some directions • Reducing the computational cost, specially in indicator-based MOEAs – are there alternatives to hypervolume? • Relaxation of the concept of dominance – cone epsilon dominance? • Surrogate-assisted many-objective optimization for expensive problems; • Co-evolutionary approaches: evolving parameters of scalar functions together with the solutions in the search space; • Visualization and decision-making tools;
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