Adaptively weighted decomposition based multi-objective evolutionary algorithm
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Adaptively weighted decomposition based multi-objective evolutionary algorithm Suraj S. Meghwani1
· Manoj Thakur2
Accepted: 22 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Multi-objective evolutionary algorithm based on Decomposition (MOEA/D) decomposes a multi-objective problem into a number of scalar optimization problems using uniformly distributed weight vectors. However, uniformly distributed weight vectors do not guarantee uniformity of solutions on approximated Pareto-Front. This study proposes an adaptive strategy to modify these scalarizing weights after regular intervals by assessing the crowdedness of solutions using crowding distance operator. Experiments carried out over several benchmark problems with complex Pareto-Fronts show that such a strategy helps in improving the convergence and diversity of solutions on approximated Pareto-Front. Proposed algorithm also shows better performance when compared with other state-of-the-art multi-objective algorithms over most of the benchmark problems. Keywords Multi-objective evolutionary algorithms · MOEA/D algorithm · Inverted generational distance · Crowding distance
1 Introduction Many complex phenomena observed in engineering, sciences, and finance are modelled as multi-objective optimization problems (MOPs). Such problems usually involve more than one conflicting objective, due to which, a vector of decision variables that simultaneously optimizes all the objectives in the concerned MOP is not realizable. Hence, the notion of optimum for MOPs is to find a set of the best trade-off between the objectives. Without the loss of generality, a general MOP can be formulated as the minimization of m objectives simultaneously. Minimize F (u) ≡ (f1 (u), f2 (u), . . . , fm (u)) u∈
(1)
where, u = (u1 , u2 , . . . , un ) is a n-dimensional vector of decision variables and is the feasible domain of these Suraj S. Meghwani
[email protected] 1
School of Computer Science and Engineering (SCOPE), Vellore Institute of Technology, Vellore, Tamil Nadu, India
2
School of Basic Sciences, Indian Institute of Technology, Mandi, Himachal Pradesh, India
variables. Let M = {1, 2, . . . , m} then for each, i ∈ M , fi : → R represents ith objective function and hence, F : ⊆ Rn → Rm . With reference to above minimization problem (1), a decision vector u is said to dominate another decision vector v, denoted by u v, if following conditions hold: 1. fi (u) ≤ fi (v), ∀i ∈ M . 2. For at-least one objective, i.e., j ∈ M , we have, fj (u) < fj (v). A solution u∗ is Pareto optimal when no other solution u ∈ dominates u∗ . A collection of all these solutions are called Pareto-optimal points denoted by PS (Pareto Set). Mathematically, it is defined as follows: PS = {u∗ ∈ | u ∈ such that u∗ ≺ u}
(2)
Consequently, another set consists of vector-valued objective function values for each of u∗ ∈ PS , denoted PF , is called Pareto-front. In general, PF may contain uncountably infinite number of points in m-dimen
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