A Convergence indicator for Multi-Objective Optimisation Algorithms

The algorithms of multi-objective optimisation had a relative growth in the last years. Thereby, it's requires some way of comparing the results of these. In this sense, performance measures play a key role. In general, it's considered some properties of these algorithms such as capacity, convergence, diversity or convergence-diversity. There are some known measures such as generational distance (GD), inverted generational distance (IGD), hypervolume (HV), Spread($\Delta$), Averaged Hausdorff distance ($\Delta_p$), R2-indicator, among others. In this paper, we focuses on proposing a new indicator to measure convergence based on the traditional formula for Shannon entropy. The main features about this measure are: 1) It does not require tho know the true Pareto set and 2) Medium computational cost when compared with Hypervolume.Comment: Submitted to TEM

Bringing Order to Special Cases of Klee's Measure Problem

Klee's Measure Problem (KMP) asks for the volume of the union of n axis-aligned boxes in d-space. Omitting logarithmic factors, the best algorithm has runtime O*(n^{d/2}) [Overmars,Yap'91]. There are faster algorithms known for several special cases: Cube-KMP (where all boxes are cubes), Unitcube-KMP (where all boxes are cubes of equal side length), Hypervolume (where all boxes share a vertex), and k-Grounded (where the projection onto the first k dimensions is a Hypervolume instance). In this paper we bring some order to these special cases by providing reductions among them. In addition to the trivial inclusions, we establish Hypervolume as the easiest of these special cases, and show that the runtimes of Unitcube-KMP and Cube-KMP are polynomially related. More importantly, we show that any algorithm for one of the special cases with runtime T(n,d) implies an algorithm for the general case with runtime T(n,2d), yielding the first non-trivial relation between KMP and its special cases. This allows to transfer W[1]-hardness of KMP to all special cases, proving that no n^{o(d)} algorithm exists for any of the special cases under reasonable complexity theoretic assumptions. Furthermore, assuming that there is no improved algorithm for the general case of KMP (no algorithm with runtime O(n^{d/2 - eps})) this reduction shows that there is no algorithm with runtime O(n^{floor(d/2)/2 - eps}) for any of the special cases. Under the same assumption we show a tight lower bound for a recent algorithm for 2-Grounded [Yildiz,Suri'12].Comment: 17 page

Approximating the least hypervolume contributor: NP-hard in general, but fast in practice

The hypervolume indicator is an increasingly popular set measure to compare the quality of two Pareto sets. The basic ingredient of most hypervolume indicator based optimization algorithms is the calculation of the hypervolume contribution of single solutions regarding a Pareto set. We show that exact calculation of the hypervolume contribution is #P-hard while its approximation is NP-hard. The same holds for the calculation of the minimal contribution. We also prove that it is NP-hard to decide whether a solution has the least hypervolume contribution. Even deciding whether the contribution of a solution is at most (1+\eps) times the minimal contribution is NP-hard. This implies that it is neither possible to efficiently find the least contributing solution (unless $P = NP$) nor to approximate it (unless $NP = BPP$). Nevertheless, in the second part of the paper we present a fast approximation algorithm for this problem. We prove that for arbitrarily given \eps,\delta>0 it calculates a solution with contribution at most (1+\eps) times the minimal contribution with probability at least $(1-\delta)$. Though it cannot run in polynomial time for all instances, it performs extremely fast on various benchmark datasets. The algorithm solves very large problem instances which are intractable for exact algorithms (e.g., 10000 solutions in 100 dimensions) within a few seconds.Comment: 22 pages, to appear in Theoretical Computer Scienc

Otimização multiobjetivo com estimação de distribuição guiada por tomada de decisão multicritério

Orientadores: Fernando José Von Zuben, Guilherme Palermo CoelhoDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de ComputaçãoResumo: Considerando as meta-heurísticas estado-da-arte para otimização multiobjetivo (MOO, do inglês Multi-Objective Optimization), como NSGA-II, NSGA-III, SPEA2 e SMS-EMOA, apenas um critério de preferência por vez é levado em conta para classificar soluções ao longo do processo de busca. Neste trabalho, alguns dos critérios de seleção adotados por esses algoritmos estado-da-arte, incluindo classe de não-dominância e contribuição para a métrica de hipervolume, são utilizados em conjunto por uma técnica de tomada de decisão multicritério (MCDM, do inglês Multi-Criteria Decision Making), mais especificamente o algoritmo TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution), responsável por ordenar todas as soluções candidatas. O algoritmo TOPSIS permite o uso de abordagens baseadas em múltiplas preferências, ao invés de apenas uma como na maioria das técnicas híbridas de MOO e MCDM. Cada preferência é tratada como um critério com uma importância relativa determinada pelo tomador de decisão. Novas soluções candidatas são então amostradas por meio de um modelo de distribuição, neste caso uma mistura de Gaussianas, obtido a partir da lista ordenada de soluções candidatas produzida pelo TOPSIS. Essencialmente, um operador de roleta é utilizado para selecionar um par de soluções candidatas de acordo com o seu mérito relativo, adequadamente determinado pelo TOPSIS, e então uma novo par de soluções candidatas é gerado a partir de perturbações Gaussianas centradas nas correspondentes soluções candidatas escolhidas. O desvio padrão das funções Gaussianas é definido em função da distância das soluções no espaço de decisão. Também foram utilizados operadores para auxiliar a busca a atingir regiões potencialmente promissoras do espaço de busca que ainda não foram mapeadas pelo modelo de distribuição. Embora houvesse outras opções, optou-se por seguir a estrutura do algoritmo NSGA-II, também adotada no algoritmo NSGA-III, como base para o método aqui proposto, denominado MOMCEDA (Multi-Objective Multi-Criteria Estimation of Distribution Algorithm). Assim, os aspectos distintos da proposta, quando comparada com o NSGA-II e o NSGA-III, são a forma como a população de soluções candidatas é ordenada e a estratégia adotada para gerar novos indivíduos. Os resultados nos problemas de teste ZDT mostram claramente que nosso método é superior aos algoritmos NSGA- II e NSGA-III, e é competitivo com outras meta-heurísticas bem estabelecidas na literatura de otimização multiobjetivo, levando em conta as métricas de convergência, hipervolume e a medida IGDAbstract: Considering the state-of-the-art meta-heuristics for multi-objective optimization (MOO), such as NSGA-II, NSGA-III, SPEA2 and SMS-EMOA, only one preference criterion at a time is considered to properly rank candidate solutions along the search process. Here, some of the preference criteria adopted by those state-of-the-art algorithms, including non-dominance level and contribution to the hypervolume, are taken together as inputs to a multi-criteria decision making (MCDM) strategy, more specifically the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), responsible for sorting all candidate solutions. The TOPSIS algorithm allows the use of multiple preference based approaches, rather than focusing on a particular one like in most hybrid algorithms composed of MOO and MCDM techniques. Here, each preference is treated as a criterion with a relative relevance to the decision maker (DM). New candidate solutions are then generated using a distribution model, in our case a Gaussian mixture model, derived from the sorted list of candidate solutions produced by TOPSIS. Essentially, a roulette wheel is used to choose a pair of the current candidate solutions according to the relative quality, suitably determined by TOPSIS, and after that a new pair of candidate solutions is generated as Gaussian perturbations centered at the corresponding parent solutions. The standard deviation of the Gaussian functions is defined in terms of the parents distance in the decision space. We also adopt refreshing operators, aiming at reaching potentially promising regions of the search space not yet mapped by the distribution model. Though other choices could have been made, we decided to follow the structural conception of the NSGA-II algorithm, also adopted in the NSGA-III algorithm, as basis for our proposal, denoted by MOMCEDA (Multi-Objective Multi-Criteria Estimation of Distribution Algorithm). Therefore, the distinctive aspects, when compared to NSGA-II and NSGA-III, are the way the current population of candidate solutions is ranked and the strategy adopted to generate new individuals. The results on ZDT benchmarks show that our method is clearly superior to NSGA-II and NSGA-III, and is competitive with other wellestablished meta-heuristics for multi-objective optimization from the literature, considering convergence to the Pareto front, hypervolume and IGD as performance metricsMestradoEngenharia de ComputaçãoMestre em Engenharia Elétrica2016/21031-0FAPESPCAPE