MOEA/D with uniformly randomly adaptive weights

Abstract

When working with decomposition-based algorithms, an appropriate set of weights might improve quality of the final solution. A set of uniformly distributed weights usually leads to well-distributed solutions on a Pareto front. However, there are two main difficulties with this approach. Firstly, it may fail depending on the problem geometry. Secondly, the population size becomes not flexible as the number of objectives increases. In this paper, we propose the MOEA/D with Uniformly Randomly Adaptive Weights (MOEA/D-URAW) which uses the Uniformly Randomly method as an approach to subproblems generation, allowing a flexible population size even when working with many objective problems. During the evolutionary process, MOEA/D-URAW adds and removes subproblems as a function of the sparsity level of the population. Moreover, instead of requiring assumptions about the Pareto front shape, our method adapts its weights to the shape of the problem during the evolutionary process. Experimental results using WFG41-48 problem classes, with different Pareto front shapes, shows that the present method presents better or equal results in 77.5% of the problems evaluated from 2 to 6 objectives when compared with state-of-the-art methods in the literature.

MOEA/D with Adaptative Number of Weight Vectors

“…One of the most popular methods for guiding the weight adaptation strategy is the Adaptive Weight Adjustment (AWA) [12,10,4,7]. A major advantage of AWA is that it changes the position of the vectors according to the feature of the MOPs and that is why we use AWA as one of the base mechanisms in this work.…”

“…A growing body of literature recognises the need to define the appropriate set of weight vectors in MOEA/D [12,10,4,7,11]. One major issue in these works is that they focus on adjusting the position of weight vectors in terms of the objective space, paying little attention to defining the number of weight vectors.…”

AREA: An adaptive reference-set based evolutionary algorithm for multiobjective optimisation

“…Associating K reference points with K solutions has a complexity Algorithm 2: Update reference set Input: population (P ), archive (A), last updated reference set (R), population size (N )); Output: Output R, B and P ; 1 Determine the number reference of points to add: K = min ( √ N , |A|); // Add K reference points 2 while |R| − N < K do 3 Compute the furthest member i of A to P by the max-min distance [32]; 4 Add i to P ; 5 Add into R the projection of i on the reference plane; // Remove K reference points 6 Calculate reference score ζ(r i ) using Eq. (4) for all r i ∈ R; 7 while |R| > N and there exists a reference score larger than 0 do // pickî (randomly if a tie exists) 8 Identify the reference with the largest score:î = argmax 1≤i≤|R| ζ(r i ); 9 Remove the reference point rî from R and the solution associating with it from P ;…”

“…A similar idea was used in the MOEA/D with adaptive weight adjustment (MOEA/D-AWA) algorithm [25], with a difference in reversing the above addition-first-removal-second order. Farias et al [8] modified MOEA/D-AWA by adopting a uniformly random method to update references every a few generations, and they reported that adaptive weight adjustment with the random method yields improved results in most of their test cases. The reference vector guided evolutionary algorithm (RVEA) [2] proposed to maintain two reference sets when handling irregular PF shapes.…”

“…The remaining constitutes a new reference set and population. Algorithm 3: Match population to reference set Input: population (P ), archive (A), reference set (R), population size (N ); Output: Updated population; 1 Merge P and A to form S, then empty P ; 2 Compute a distance matrix E between pairs of R and S; 3 while |P | < N do 4 Identify the nearest reference point for each member of S, and the save them in a set I; 5 for each reference point i ∈ I do 6 Select the nearest individual sk in S to i, and associate it with i; 7 Save sk to P ; 8 Set to infinity all the elements of the i-th row andk-th column of the distance matrix E;…”