Prior to the introduction, we recommend three good many-objective optimizers:
(i) PICEA-g from preference-based coevolutionary class
(ii) HypE from indicator based class
(iii) NSGA-III from decomposition based class
Introduction to MOPs, EMO, and MaOPs
Multi-objective problems (MOPs) regularly arise in real-world design scenarios, where two or more objectives are required to be optimized simultaneously. As such objectives are often in competition with one another, the optimal solution of MOPs is a set of trade-off solutions, rather than a single solution. Due to the population-based approach, multi-objective evolutionary algorithms (MOEAs) are well suited for solving MOPs since this leads naturally to the generation of an approximate trade-o_ surface (or Pareto front) in a run. So far there are mainly three classes of MOEAs according to the selection strategies: Pareto dominance based ones, decomposition based ones and indicator based ones.
Pareto-dominance based MOEAs, e.g., MOGA (Fonseca and Fleming 1993), NSGA-II (Deb
et al. 2002) and SPEA2 (Zitzler et al. 2002), were some of the earliest approaches and are generally accepted to perform well on MOPs with 2 and 3 objectives. However, their search capability often degrades significantly as the number of objectives increases (Purshouse and Fleming 2003). This is because the proportion of Pareto optimal (or non-dominated) objective vectors in the population grows large when MOPs have more than 3 objectives i.e. many-objective problems (MaOPs). As a result, insufficient selection pressure is generated towards the Pareto front.
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Recently, Purshouse et al. (2011) proposed a novel concept for addressing MaOPs; co-evolution of a set of decision-maker preferences with a population of candidate solutions is introduced during the search. Candidate solutions are guided by the co-evolved preferences towards the Pareto optimal front. Two instantiations of this concept have been realised: a preference-inspired coevolutionary algorithm using goal vectors (PICEA-g) and a preference-inspired co-evolutionary algorithm using weight vectors (PICEA-w). Both PICEAs perform well. PICEA-g is demonstrated to outperform "epsilon-MOEA, average ranking based MOEA, HypE, MOEA/D and NSGA-II on MaOPs (Purshouse et al. 2011, Wang et al. 2013a). PICEA-w, as a decomposition based MOEA, outperforms some state-of-the-art decomposition based algorithms such as MOGLS, MOEA/D, MSOPS, and EMOSA on MaOPs with different Pareto front geometries (Wang et al. 2013b, 2014).
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Source Code: PICEA-g (Matlab) Preference-inspired Coevolutionary Algorithm using goal vectors, TEVC Many-objective optimizer (SBX and PM as genetic operators)
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Source Code: PICEA-g (C++) ( DE as genetic operator)
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Source Code: MOEA/D-LPBI (Matlab) On the effect of LPBI in MOEA/D for multi objective optimization, IEEE SSCI paper with Hisao Ishibuchi et al.
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Source Code: MOEA/D-LWS (Matlab) Localised weighted sum method for many objective optimization, TEVC paper with Hisao Ishibuchi et al
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Source Code: MOEA/D-PaS (Matlab) Decomposition Based Algorithms Using Pareto Adaptive Scalarizing Method, TEVC paper with Qingfu Zhang et al
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Source Code: MOEA/D-PaP (Matlab) Pareto Adaptive Penalty based Boundary Intersection method for multi objective optimization, Information Sciences paper, 2017
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Source Code: MOEA/D-RP (Matlab) On the effect of reference point in MOEA/D for multi-objective optimization
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Source Code: PICEA-ng-for-HRES (Matlab) Optimal design of hybrid renewable energy systems using multi-objective evolutionary algorithm
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Source Code: Extended WFG (Matlab) An extended wfg test problems where test problems with various Pareto optimal front shapes are provided
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Source Code: Performace metric of Attainment surface (Matlab)
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