Recent Results and Open Problems in Evolutionary Multiobjective - - PowerPoint PPT Presentation

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

Recent Results and Open Problems in Evolutionary Multiobjective Optimization

Carlos A. Coello Coello

CINVESTAV-IPN Evolutionary Computation Group (EVOCINV) Departamento de Computaci´

• n
• Av. IPN No. 2508, Col. San Pedro Zacatenco

M´ exico, D.F . 07360, MEXICO ccoello@cs.cinvestav.mx

Wellington, New Zealand

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

Outline

1

Introduction

2

Recent Results and Open Problems Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

3

The Challenges of this Century

4

Conclusions

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

Multi-Objective Evolutionary Algorithms

Evolutionary algorithms seem particularly suitable to solve multiobjective optimization problems, because they deal simultaneously with a set of possible solutions (the so-called population). This allows us to find several members of the Pareto optimal set in a single run of the algorithm, instead of having to perform a series of separate runs as in the case of the traditional mathematical programming techniques. Additionally, evolutionary algorithms are less susceptible to the shape or continuity of the Pareto front (e.g., they can easily deal with discontinuous or concave Pareto fronts), whereas these two issues are normally a real concern for mathematical programming techniques.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

Historical Highlights

The potential of evolutionary algorithms in multiobjective

• ptimization was hinted by Richard S. Rosenberg in his PhD

thesis from 1967, which is entitled “Simulation of genetic populations with biochemical properties”.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

Historical Highlights

However, the first actual implementation of a multi-objective evolutionary algorithm was John David Schaffer’s Vector Evaluated Genetic Algorithm (VEGA), which dates back to

• 1984. This was a naive multi-objective evolutionary algorithm,

whose selection mechanism did not rely on Pareto optimality.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

Historical Highlights

First Generation From 1985 to 1998, we can identify a first generation of multi-objective evolutionary algorithms (MOEAs) which were non-elitist and relatively naive: Vector Evaluated Genetic Algorithm (VEGA) (1985) Nondominated Sorting Genetic Algorithm (NSGA) (1994) Niched-Pareto Genetic Algorithm (NPGA) (1994) Multi-Objective Genetic Algorithm (MOGA) (1993)

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

Historical Highlights

Second Generation The second generation brought us a set of elitist MOEAs which were more efficient and effective and had a more elegant design: SPEA and SPEA2 (1999, 2001) NSGA-II (2000,2002) micro-GA for MOO (2001) PAES (2000) PESA, PESA-II (2000, 2001) Many others ...

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

Historical Highlights

Since Goldberg’s early proposal (1989), MOEAs consist of two basic components: A selection mechanism that normally (but not necessarily) incorporates Pareto optimality. A density estimator, which is responsible for maintaining diversity, and therefore, keeping the MOEA from converging to a single solution.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

Historical Highlights

The main density estimators that have been used with MOEAs are: Fitness sharing Clustering Entropy Adaptive grids Crowding

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

Number of papers published per year (up to mid 2015)

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Introduction After 30 years of existence, and with so much work done, EMO may seem intimidating to some people. If so many people have worked in this area for the last 15 years, what remains to be done?

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Luckily, there still are many opportunities to do research in this area, even within topics that seem to have been visited a lot in the past. Imagination is more important than knowledge. Albert Einstein

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Algorithms The main current research trend regarding algorithmic development is to adopt a performance measure in the selection scheme of a MOEA. See for example: ESP: The Evolution Strategy with Probability Mutation uses a hypervolume-based, scaling independent, parameterless measure, to truncate overpopulated external archives (Huband et al., 2003). IBEA: This is a framework that allows any performance indicator to be incorporated into the selection mechanism

• f a MOEA (Zitzler et al., 2004). Its authors tested it with

the hypervolume and with the binary ǫ indicator.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Algorithms SMS-EMOA: The S Metric Selection Evolutionary Multiobjective Algorithm is based on the hypervolume performance measure (Emmerich et al., 2005; Beume et al., 2007). SPAM: The Set Preference Algorithm for Multiobjective

• ptimization is meant to generalize IBEA by allowing any

sort of set preference relation (Zitzler et al., 2008).

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Why to use Hypervolume? The Hypervolume (also known as the S metric or the Lebesgue Measure) of a set of solutions measures the size of the portion of objective space that is dominated by those solutions collectively. Its Good Side Advantage: It has been proved that the maximization of this performance measure is equivalent to finding the Pareto optimal set (Fleischer, 2003).

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Its Good Side Advantage: Empirical studies have shown that (for a certain number of points previously determined) the maximization of the hypervolume does indeed produce subsets of the Pareto front which are well-distributed (Knowles, 2003; Emmerich, 2005). Advantage: It measures convergence and, to a certain extent, also the spread of solutions along the Pareto front.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Its Bad Side Disadvantage: The computation of this performance measure depends of a reference point, which can influence the results in a significant manner. Some people have proposed to use the worst objective function values in the current population, but this requires scaling of the

• bjectives.

Disadvantage: The best algorithms known to compute hypervolume have a polynomial complexity on the number

• f points used, but such complexity grows exponentially on

the number of objectives!

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Probably, we should look at other performance measures as possible alternatives to the hypervolume. For example, the binary indicators proposed by Hansen and Jaszkiewicz (1998). In fact, there is a recent performance measure proposed by Sch¨ utze et al. (2011), which is based on the Hausdorff distance, and which can be used in the selection mechanism of a MOEA. This sort of selection mechanism seems to scale well with the number of objectives, it’s computationally inexpensive and seems to provide results similar to those obtained by hypervolume-based MOEAs (see (Rodr´ ıguez and Coello, 2012)).

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

It is also possible to estimate the hypervolume (using, for example, Monte Carlo simulations) in order to produce a more efficient hypervolume-based MOEA (see for example: Hype (Bader & Zitzler, 2011)). There are also other performance measures that could be used for selection but have not been explored (e.g., the inverted generational distance).

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Another interesting idea is the design of MOEAs based on scalarization methods. Here, the main approach has been MOEA/D (a MOEA based on decomposition) (Zhang, 2007). This is a very old idea (which is based on the Normal Boundary Intersection method, proposed in the 1990s), which has given rise to powerful MOEAs which, however, may be a bit impractical in the presence of too many objectives.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Recently, the use of the R2 indicator has also attracted a lot of interest because of its nice properties (Brockhoff et al., 2012), and several MOEAs based on this indicator are already available (see for example (Phan and Suzuki, 2013); (Hern´ andez-G´

• mez and Coello Coello, 2013); (D´

ıaz-Manr´ ıquez et al., 2013)).

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

MOEAs that few people know about Many other MOEAs have been proposed, but have been rarely used: The Nash Genetic Algorithm (Sefrioui, 1996). The Maximin Fitness Function (Balling, 2000). Incrementing Multi-Objective Evolutionary Algorithm (IMOEA) (Tan et al., 2001). Constraint Method-Based Evolutionary Algorithm (CMEA) for Multiobjective Optimization (Ranjithan et al., 2001). Orthogonal Multi-Objective Evolutionary Algorithm (OMOEA) (Zeng et al., 2004).

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

MOEAs for Expensive Objective Functions Many real-world problems have objective functions which are very expensive to evaluate (e.g., in aeronautical engineering). Few MOEAs currently exist to deal with such problems.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

MOEAs for Expensive Objective Functions Currently, there are three main lines of algorithmic design to deal with these problems:

1

Use of “clever” approaches to save objective function

• evaluations. The most common sort of approach is the use
• f fitness approximation schemes.

2

Use of surrogate methods, which adopt an approximation of the objective function which is cheap to evaluate and then adjust the error of the model by performing few evaluations

• f the real objective function(s).

3

Use of parallel MOEAs. Although you can get very creative here, few people have actually proposed interesting approaches.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Fitness Approximation The idea here is to estimate the quality of some of the individuals based on an approximation model of the fitness

• landscape. Thus, for the single-objective case, fitness

approximation schemes estimate the fitness of an individual based on the previously observed objective function values of its neighboring individuals.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Fitness Approximation The schemes can go from very simple ideas, such as fitness inheritance in which an offspring’s fitness is estimated by using the weighted average fitness of its parents, to the use of neural networks or statistical models built from a few points that are adopted to predict the fitness of new individuals. Taking this to the multiobjective case is not trivial, and is currently an interesting line of research.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Surrogate Methods There is a lot of work done on surrogate methods in the engineering literature. Even for the multiobjective case, researchers have tested approaches such as radial basis functions, kriging, regression models, among others. Normally, kriging is considered to be the best choice, if we can afford its high computational cost. The implementation of MOEAs that incorporate surrogate methods is not trivial!

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

ParEGO It’s probably the best known MOEA based on surrogate methods (at least for the EMO community). It requires only between 100 and 250 objective function evaluations to produce reasonably good approximations in problems of low dimensionality (up to 10 decision variables).

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Parallel MOEAs Although parallel EAs have been relatively popular in the literature, relatively few research exists on designing parallel MOEAs that really exploit parallel architectures. A number of interesting issues arise when designing parallel MOEAs. For example: should we use local external archives or only a global

• ne? Is it possible to explore non-overlapping portions of the

search space with each deme? What about GPU-based implementations (can we do this in an efficient and effective way)?

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Parallel MOEAs A few interesting parallel MOEAs currently exist: The Predator-Prey Model: It places solutions (preys) on the vertices of an undirected connected graph, defining a

• neighborhood. Such preys are caught by predators that

perform random walks within a certain neighborhood. Predators are only interested in preys that are good in a particular objective. Thus, preys that are good in all the

• bjectives are given a higher chance of survival and can

produce more descendants. An interesting approach that shares similarities with the predator-prey model is the cellular multi-objective genetic algorithm proposed by Alba et al. (2003).

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

DRMOGA: The Divided Range Multi-Objective Genetic Algorithm (Hiroyasu, 2000) was one of the earliest attempts to divide the effort of the search done by a parallel MOEA in a clever way. Zhu (2002) and Deb (2003) proposed other approaches that also aimed to divide the search effort of a parallel MOEA. MRMOGA: This approach uses an island model in which migration is unidirectional, and is done in such a way that the resolution gets finer as we move to a new deme (Jaimes & Coello, 2005).

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

MOEAs for Expensive Objective Functions There are, of course, other possible ideas to deal with expensive functions: In my research group, we have developed MOEAs based

• n differential evolution and particle swarm optimization

that can solve the ZDT test problems and several of the DTLZ test problems, with less than 4,000 objective function evaluations without using surrogates. The main idea of these approaches is to increase a lot the selection pressure at the expense of sacrificing diversity. Then, using a handful of points located on the Pareto front (or very close to it), we can rebuild the rest of the front by using powerful local search engines (e.g., rough sets).

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

MOEAs for Expensive Objective Functions Other ideas: Hybridization of MOEAs with gradient-based methods. There are a lot of interesting issues here, but if the hybridization is cleverly done, very efficient algorithms can be designed (e.g., Hernandez-Diaz et al., 2008, Lara et al., 2010). We could also hybridize MOEAs with direct search methods (e.g., Nelder and Mead) so that we don’t have to estimate the derivatives (see for example, Zapotecas & Coello, 2008).

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Self-Adaptation and Online Adaptation Whatever happened with self-adaptation (and online adaptation) and the dream of creating a parameterless multi-objective evolutionary algorithm? After a few isolated efforts, such as the mutation operator based on Kohonen’s self-organizing maps (B¨ uche, 2002), the adaptive parameters

• f IMOEA (Tan et al., 2001) and the parameterless microGA2

(Toscano & Coello, 2003), not much work has been done in this regard.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Self-Adaptation and Online Adaptation The main problem for designing self-adaptive (or online adaptive) MOEAs lies on the difficulty to define convergence criteria, as well as the population behavior that can be used to guide the search in a proper manner. Although performance measures and Pareto optimality can be used for these tasks, it is evident that the parameters setting of a MOEA is much more difficult than that of a single-objective evolutionary algorithm.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Stopping Criteria Another missing mechanism to get to a parameterless MOEA is a well-established stopping criterion. There have been some attempts in this direction: The rank-histograms from Kumar & Rockett (1997,2002). The micro-GA2 (Toscano & Coello, 2003) stops when the solutions generated cannot be improved after a certain number of iterations.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Stopping Criteria Mart´ ı et al. (2007) proposed a mechanism that gathers information about the solutions obtained so far. This information is accumulated and updated using a discrete Kalman filter. This accumulated information is used to decide when to stop a MOEA. Trautmann et al. (2008,2009) proposed a convergence criterion based on statistical testing. There is also an interesting survey on this topic that was presented at EMO’2011 (Wagner et al., 2011).

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Self-Adaptation and Online Adaptation The other problem with self-adaptation is its extra computational cost (i.e., the additional objective function evaluations that it requires), which may be unaffordable in certain applications. Self-Adaptation and Online Adaptation Another important issue is the definition of the parameters that are worth self-adapting (e.g., mutation and crossover rates, population size, and even the encoding to be used).

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Many-Objectivity MOEAs that adopt a selection mechanism based on Pareto

• ptimality do not scale properly: the number of nondominated

solutions grows exponentially with the number of objectives (Farina, 2004). This makes the selection mechanism completely useless, since all the nondominated solutions are considered equally good!

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

It has been empirically shown that a random search is more effective than the NSGA-II when dealing with more than 10

• bjectives (Mostaghim & Schmeck, 2008).

There is also another interesting problem related to scalability: as we increase the number of objectives, the number of solutions required to sample the Pareto front, also grows exponentially.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Many-Objective Problems In the current literature we can identify two approaches commonly adopted to cope with many-objective problems: Adopt or propose a preference relation that induces a finer grain order on the solutions than that induced by the Pareto dominance relation (Di Pierro, 2007; Farina, 2002; S¨ ulflow, 2007; Sato, 2007). To reduce the number of objectives of the problem during the search process (Brockhoff, 2006), or a posteriori, during the decision making process (Deb, 2006; Brockhoff, 2007; Lopez Jaimes, 2008).

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Alternative Preference Relations Average Ranking Method: This method computes for each solution a different rank considering each objective

• independently. The final rank of a solution is obtained by

summing all theirs ranks on each objective (Bentley, 1997). k-Optimality: It’s a relaxed form of Pareto dominance that takes into account the number of improved objectives between two solutions (Farina & Amato, 2004).

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Alternative Preference Relations Preference Order Ranking: A point x∗ is efficient in order k if f(x∗) is not dominated by any point for any of the k-element subsets of the objectives. The condition of efficiency of order can be used to help reduce the number

• f points in a set by retaining only those that are regarded

as “best compromises” (di Pierro, 2006).

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Alternative Preference Relations Control of the dominance area: Sato et al. (2007) proposed a method to control the dominance area of

• solutions. This method can control the degree of

expansion or contraction of the dominance area by adopting a user-defined parameter.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Alternative Preference Relations Favour Relation: It consists of a new relation called

• favour. This technique requires no user interaction and can

handle infeasible solutions. x is favoured to y (x <f y) iff i components of x are better than the corresponding components of y and only j components of y are better than the corresponding components of x.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Dimensionality Reduction The main idea is to identify the redundant objectives (or redundant to some degree) in order to discard them. A redundat objective is one that can be removed without changing the dominance relation. Deb and Saxena (2005) proposed a method for reducing the number of objectives based on principal component analysis. The main assumption is that if two objectives are negatively correlated (taking the generated Pareto front as the data set), then these objectives are in conflict with each other.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Dimensionality Reduction Brockhoff and Zitzler (2006) defined two kinds of objective reduction problems and two corresponding algorithms to solve them. Here, the conflict is defined using the change in the dominance relation induced by the set of objectives

• ver a solution set in the objective space. That is, if the

dominance relation among the vectors does not change when an objective is discarded, then that objective is not in conflict with the other objectives and therefore is considered as redundant.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Other approaches are, of course possible: Use of linear or nonlinear aggregating functions. Use of alternative schemes (e.g., adopting machine learning techniques, such as in the multiobjective neural estimation of distribution algorithm (MONEDA) (Mart´ ı et al., 2008)). Use of alternative archiving techniques (e.g., the Two-Archive MOEA, which uses one archive for convergence and another for diversity (Praditwong & Yao, 2006)).

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

What about scalability in decision variable space? There is almost no work on this topic, which also deserves

• attention. Zhang & Lim (2007) performed a small study with a

single problem that was scaled up to 100 decision variables. Durillo et al. (2008) did a more thorough study, adopting from 8 to 2048 decision variables. Perhaps the most remarkable finding was that PAES was the most salient technique from the several compared (NSGA-II, SPEA2, MOCell, OMOPSO and PESA-II). OMOPSO did very well up to 256 decision variables and ranked second between 512 and 1024 decision variables.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions Algorithms MOEAs for Expensive Objective Functions Self-Adaptation and Online Adaptation Scalability

Recent Results and Open Problems

Recently, the first MOEA explicitly designed for large-scale problems was introduced (Antonio and Coello Coello, 2013). This MOEA is based on a coevolutionary approach and can deal with problems having up to 5,000 decision variables.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

The Future

The Challenges of this Century After 30 years of activity, from which about 15 have been quite intense, we can say that this area has lost its innocence. Today, it’s not enough to design a slight variation of an existing algorithm to get it published in a specialized journal. We can say that the age of the straightforward problems is over. We now experience the growing pains, since we are in transition towards an age in which we start seeing a lot of work “by analogy” and few novel ideas.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

The Future

The Challenges of this Century However, several problems remain that are worth studying, although the solution of most of them may consume a considerable amount of time. This has made the discipline less friendly than before and may scare away those who got here

• ut of curiosity. Let’s see if these problems (some of which are

briefly described next) are studied in sufficient depth in the years to come.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

The Future

The Challenges of this Century Sources of Difficulty: What makes a problem difficult for a MOEA? Clearly, this is a fundamental question for this discipline, but today we have little information in this

• regard. We know, for example, that a disconnection in

decision variable space causes more trouble than a disconnection in objective function space.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

The Future

The Challenges of This Century However, we do not know what features are desirable in a MOEA so that it can efficiently solve real-world problems (or at least a particular subclass). Current test problems are much more challenging than those existing 10 years ago, but that provides no indication regarding the suitability

• f today’s MOEAs for solving real-world problems.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

The Future

The Challenges of This Century Efficiency: What is the efficiency limit that we can reach with a MOEA? Today, there are MOEAs that require less than 5,000 objective function evaluations to generate reasonably good approximations of the Pareto front of problems with 10 or more decision variables. Can we reduce this figure to 1,000? Can we design such a MOEA with a “robust” behavior? What is robustness in a multi-objective context?

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

The Future

The Challenges of This Century Constraint-Handling: In its origins, evolutionary multi-objective optimization did not pay much attention to

• constraints. Over the years, simple constraint-handling

methods based on straightforward modifications to the Pareto dominance rules were adopted. As of today, very few approaches exist to explicitly handle constraints in multi-objective optimization problems.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

The Future

The Challenges of This Century Ironically, multi-objective optimization concepts have inspired constraint-handling techniques used for single-objective optimization (see for example [Coello, 2000; Yen, 2003]). However, the desdain for constrained problems is reflected even in the current benchmarks adopted for testing new MOEAs.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

The Future

The Challenges of This Century Back to our Roots: I’m firmly convinced that we have considerably underestimated the potential of hybridizing evolutionary algorithms with mathematical programming

• techniques. There used to be algorithms based on game

theory, min-max optimality, and the ε-constraint method, but many researchers considered them “politically incorrect”. In recent years, however, several researchers have emphasized the advantages of hybridizing MOEAs with mathematical programming techniques (see for example (Shukla, 2007; Lara et al., 2010; Zapotecas & Coello, 2010; Bosman, 2012)).

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

The Future

The Challenges of This Century Uncertainty and Dynamism: Topics such as stochastic dominance or stochastic multi-objective combinatorial problems, have been scarcely dealt with in the specialized literature and can give rise to new MOEAs. The same can be said about dynamic problems, although there is more work done in this topic (see for example (Farina & Deb, 2004)).

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

The Future

The Challenges of This Century Visualization: Another interesting topic is the visualization

• f a Pareto front in a high-dimensional space. There are

some recent proposals in this regard (e.g., Pryke et al., 2006; Obayashi & Sasaki, 2003). There is also some interesting work done in the MCDM community in this regard (Lotov, 2004).

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

The Future

The Challenges of This Century Test Problems: The ZDT and the DTLZ test problems have been widely used, although there are several other benchmarks which have been less popular. We have, for example, Okabe’s problems (Okabe, 2004) and the WFG test problems from Huband et al. (2005 and 2006), which are among the hardest proposed so far. However, these test problems, regardless of how difficult they are, are loosely related to the actual difficulties that real-world problems have.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

The Future

The Challenges of This Century The Third Generation: One day, somebody will have an idea sufficiently powerful for others to follow him/her. This will probably be an algorithm, although it could be simply an operator or a specific mechanism. I wonder how big will this community be by then.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

Conclusions

Evolutionary multi-objective optimization still has a lot to offer regarding research work. However, the new age that we are currently living in this discipline, requires more commitment and the generation of more profound ideas. Probably some simple problems still remain, but we have to look for them more

• carefully. Clearly, the design of algorithms tailored for a

particular problem (of a high degree of difficulty) will remain as an active research area during a long time.

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

To know more about evolutionary multi-objective optimization

Please visit our EMOO repository located at: http://delta.cs.cinvestav.mx/˜ccoello/EMOO with a mirror at: http://www.lania.mx/˜ccoello/EMOO

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

To know more about evolutionary multi-objective optimization

Carlos A. Coello Coello Recent Results and Open Problems in EMO

Contents Introduction Recent Results and Open Problems The Challenges of this Century Conclusions

To know more about evolutionary multi-objective optimization

The EMOO repository currently contains: Over 9870 bibliographic references including 286 PhD theses, 50 Masters theses, over 4470 journal papers and

• ver 3710 conference papers.

Contact information of 79 EMOO researchers. Public domain implementations of SPEA, NSGA, NSGA-II, the microGA, MOGA, ǫ-MOEA, MOPSO and PAES, among

• thers.

Carlos A. Coello Coello Recent Results and Open Problems in EMO