In dealing with constrained multi-objective optimization problems (CMOPs), a key issue of multi-objective evolutionary algorithms (MOEAs) is to balance the convergence and diversity of working populations.
Employing a number of sub-populations to solve problems in a collaborative way @cite_26 is a widely used approach, which can help an algorithm balance its convergence and diversity. One of the most popular methods is the M2M population decomposition approach @cite_41 , which decomposes a multi-objective optimization problems into a number of simple multi-objective optimization subproblems in the initialization, then solves these sub-problems simultaneously in a coordinated manner. For this purpose, @math unit vectors @math in @math are chosen in the first octant of the objective space. Then @math is divided into @math subregions @math , where @math is where @math is the acute angle between @math and @math . Therefore, the population is decomposed into @math sub-populations, each sub-population searches for a different multi-objective subproblem. Subproblem @math is defined as:
@cite_41 @cite_26
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This letter suggests an approach for decomposing a multiobjective optimization problem (MOP) into a set of simple multiobjective optimization subproblems. Using this approach, it proposes MOEA D-M2M, a new version of multiobjective optimization evolutionary algorithm-based decomposition. This proposed algorithm solves these subproblems in a collaborative way. Each subproblem has its own population and receives computational effort at each generation. In such a way, population diversity can be maintained, which is critical for solving some MOPs. Experimental studies have been conducted to compare MOEA D-M2M with classic MOEA D and NSGA-II. This letter argues that population diversity is more important than convergence in multiobjective evolutionary algorithms for dealing with some MOPs. It also explains why MOEA D-M2M performs better. This paper introduces a parallel and distributed algorithm for solving the following minimization problem with linear constraints: @math minimizef1(x1)+ź+fN(xN)subject toA1x1+ź+ANxN=c,x1źX1,ź,xNźXN,where @math Nź2, @math fi are convex functions, @math Ai are matrices, and @math Xi are feasible sets for variable @math xi. Our algorithm extends the alternating direction method of multipliers (ADMM) and decomposes the original problem into N smaller subproblems and solves them in parallel at each iteration. This paper shows that the classic ADMM can be extended to the N-block Jacobi fashion and preserve convergence in the following two cases: (i) matrices @math Ai are mutually near-orthogonal and have full column-rank, or (ii) proximal terms are added to the N subproblems (but without any assumption on matrices @math Ai). In the latter case, certain proximal terms can let the subproblem be solved in more flexible and efficient ways. We show that @math źxk+1-xkźM2 converges at a rate of o(1 k) where M is a symmetric positive semi-definte matrix. Since the parameters used in the convergence analysis are conservative, we introduce a strategy for automatically tuning the parameters to substantially accelerate our algorithm in practice. We implemented our algorithm (for the case ii above) on Amazon EC2 and tested it on basis pursuit problems with >300 GB of distributed data. This is the first time that successfully solving a compressive sensing problem of such a large scale is reported. Many-objective problems (MAPs) have put forward a number of challenges to classical Pareto-dominance based multi-objective evolutionary algorithms (MOEAs) for the past few years. Recently, researchers have suggested that MOEA D (multi-objective evolutionary algorithm based on decomposition) can work for MAPs. However, there exist two difficulties in applying MOEA D to solve MAPs directly. One is that the number of constructed weight vectors is not arbitrary and the weight vectors are mainly distributed on the boundary of weight space for MAPs. The other is that the relationship between the optimal solution of subproblem and its weight vector is nonlinear for the Tchebycheff decomposition approach used by MOEA D. To deal with these two difficulties, we propose an improved MOEA D with uniform decomposition measurement and the modified Tchebycheff decomposition approach (MOEA D-UDM) in this paper. Firstly, a novel weight vectors initialization method based on the uniform decomposition measurement is introduced to obtain uniform weight vectors in any amount, which is one of great merits to use our proposed algorithm. The modified Tchebycheff decomposition approach, instead of the Tchebycheff decomposition approach, is used in MOEA D-UDM to alleviate the inconsistency between the weight vector of subproblem and the direction of its optimal solution in the Tchebycheff decomposition approach. The proposed MOEA D-UDM is compared with two state-of-the-art MOEAs, namely MOEA D and UMOEA D on a number of MAPs. Experimental results suggest that the proposed MOEA D-UDM outperforms or performs similarly to the other compared algorithms in terms of hypervolume and inverted generational distance metrics on different types of problems. The effects of uniform weight vector initializing method and the modified Tchebycheff decomposition are also studied separately. Decomposition-based methods are an increasingly popular choice for a posteriori multi-objective optimization. However the ability of such methods to describe a trade-off surface depends on the choice of weighting vectors defining the set of subproblems to be solved. Recent adaptive approaches have sought to progressively modify the weighting vectors to obtain a desirable distribution of solutions. This paper argues that adaptation imposes a non-negligible cost -- in terms of convergence -- on decomposition-based algorithms. To test this hypothesis, the process of adaptation is abstracted and then subjected to experimentation on established problems involving between three and 11 conflicting objectives. The results show that adaptive approaches require longer traversals through objectivespace than fixed-weight approaches. Since fixed weights cannot, in general, be specified in advance, it is concluded that the new wave of decomposition-based methods offer no immediate panacea to the well-known conflict between convergence and distribution afflicting Pareto-based a posteriori methods.