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A multiobjective evolutionary algorithm based on decomposition
 IEEE Transactions on Evolutionary Computation, Accepted
, 2007
"... 1 Decomposition is a basic strategy in traditional multiobjective optimization. However, this strategy has not yet widely used in multiobjective evolutionary optimization. This paper proposes a multiobjective evolutionary algorithm based on decomposition (MOEA/D). It decomposes a MOP into a number o ..."
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Cited by 44 (14 self)
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1 Decomposition is a basic strategy in traditional multiobjective optimization. However, this strategy has not yet widely used in multiobjective evolutionary optimization. This paper proposes a multiobjective evolutionary algorithm based on decomposition (MOEA/D). It decomposes a MOP into a number of scalar optimization subproblems and optimizes them simultaneously. Each subproblem is optimized by using information from its several neighboring subproblems, which makes MOEA/D have lower computational complexity at each generation than MOGLS and NSGAII. Experimental results show that it outperforms or performs similarly to MOGLS and NSGAII on multiobjective 01 knapsack problems and continuous multiobjective optimization problems. Index Terms multiobjective optimization, decomposition, evolutionary algorithms, memetic algorithms, Pareto optimality, computational complexity. I.
Optimality conditions for vector optimization problems with variable ordering structures
, 2010
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EQUISPACED PARETO FRONT CONSTRUCTION FOR CONSTRAINED BIOBJECTIVE OPTIMIZATION ∗
"... Abstract. We consider constrained biobjective optimization problems. One of the extant issues in this area is that of uniform sampling of the Pareto front. We utilize equispacing constraints on the vector of objective values, as discussed in a previous paper dealing with the unconstrained problem. W ..."
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Abstract. We consider constrained biobjective optimization problems. One of the extant issues in this area is that of uniform sampling of the Pareto front. We utilize equispacing constraints on the vector of objective values, as discussed in a previous paper dealing with the unconstrained problem. We present a direct and a dual formulation based on arclength homotopy continuation and illustrate the direct method (using standard nonlinear programming tools) on some problems from the literature. We contrast the performance of our method with the results of three other algorithms, showing several orders of magnitude speedup with respect to evolutionary algorithms, while simultaneously providing perfectly sampled fronts by construction. We then consider a largescale application: the variational approach to mesh generation for partial differential equations in complex domains. Balancing multiple criteria leads to significantly improved mesh design.
Approximating the Set of Pareto Optimal Solutions in Both the Decision and Objective Spaces by an Estimation of Distribution Algorithm
, 2008
"... Most existing multiobjective evolutionary algorithms aim at approximating the PF, the distribution of the Pareto optimal solutions in the objective space. In many reallife applications, however, a good approximation to the PS, the distribution of the Pareto optimal solutions in the decision space, ..."
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Cited by 3 (0 self)
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Most existing multiobjective evolutionary algorithms aim at approximating the PF, the distribution of the Pareto optimal solutions in the objective space. In many reallife applications, however, a good approximation to the PS, the distribution of the Pareto optimal solutions in the decision space, is also required by a decision maker. This paper considers a class of MOPs, in which the dimensionalities of the PS and PF are different so that a good approximation to the PF might not approximate the PS very well. It proposes a probabilistic model based multiobjective evolutionary algorithm, called MMEA, for approximating the PS and the PF simultaneously for a MOP in this class. In the modelling phase of MMEA, the population is clustered into a number of subpopulations based on their distribution in the objective space, the PCA technique is used to detect the dimensionality of the centroid of each subpopulation, and then a probabilistic model is built for modelling the distribution of the Pareto optimal solutions in the decision space. Such modelling procedure could promote the population diversity in both the decision and objective spaces. To ease the burden of setting the number of subpopulations, a dynamic strategy for periodically adjusting it has been adopted in MMEA. The experimental comparison between MMEA and the two other methods, KP1 and OmniOptimizer on a set of test instances, some of which are proposed in this paper, have been made in this paper. It is clear from the experiments that MMEA has a big advantage over the two other methods in approximating both the PS and the PF of a MOP when the PS is a nonlinear manifold, although it might not be able to perform significantly better in the case when the PS is a linear manifold. Index Terms Multiobjective optimization, Pareto optimality, estimation of distribution algorithm, principal component analysis.
Multiobjective Problems of Convex Geometry
 Siberian Math. J
, 2009
"... Abstract: Under study is the new class of geometrical extremal problems in which it is required to achieve the best result in the presence of conflicting goals; e.g., given the surface area of a convex body x, we try to maximize the volume of x and minimize the width of x simultaneously. These prob ..."
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Cited by 3 (2 self)
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Abstract: Under study is the new class of geometrical extremal problems in which it is required to achieve the best result in the presence of conflicting goals; e.g., given the surface area of a convex body x, we try to maximize the volume of x and minimize the width of x simultaneously. These problems are addressed along the lines of multiple criteria decision making. We describe the Paretooptimal solutions of isoperimetrictype vector optimization problems on using the techniques of the space of convex sets, linear majorization, and mixed volumes.
Scalarizations for adaptively solving multiobjective optimization problems
, 2006
"... In this paper several parameter dependent scalarization approaches for solving nonlinear multiobjective optimization problems are discussed and it is shown that they can be considered as special cases of a scalarization problem by Pascoletti and Serafini (or a modification of this problem). Based o ..."
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In this paper several parameter dependent scalarization approaches for solving nonlinear multiobjective optimization problems are discussed and it is shown that they can be considered as special cases of a scalarization problem by Pascoletti and Serafini (or a modification of this problem). Based on these connections theoretical results as well as a new algorithm for adaptively controlling the choice of the parameters for generating almost equidistant approximations lately developed for the PascolettiSerafini scalarization can be applied to these problems. For example for such wellknown problems as the econstraint or the normal boundary intersection method algorithms for adaptively generating high quality approximations are derived.
Quality Discrete Representations in Multiple Objective Programming
"... Within the past ten years, more emphasis has been placed on generating discrete representations of the nondominated set which are truly representative of the nondominated set as a whole. This paper reviews measures for assessing the quality of discrete representations as well as exact solution meth ..."
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Within the past ten years, more emphasis has been placed on generating discrete representations of the nondominated set which are truly representative of the nondominated set as a whole. This paper reviews measures for assessing the quality of discrete representations as well as exact solution methods that attempt to produce representations satisfying certain quality criteria. The measures are classified according to the aspect of the representation which they assess: cardinality, coverage, or spacing. The proposed solution methods are categorized according to whether a measure is integrated into the procedure a priori (before generation of solution points), a posteriori (after the generation of solution points), or not at all. The paper concludes with a comparative discussion of these three approaches and directions for future research.
Ordering Structures in Vector Optimization and Applications in Medical Engineering
, 2013
"... This manuscript is on the theory and numerical procedures of vector optimization w.r.t. various ordering structures, on recent developments in this area and, most important, on their application to medical engineering. In vector optimization one considers optimization problems with a vectorvalued ..."
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This manuscript is on the theory and numerical procedures of vector optimization w.r.t. various ordering structures, on recent developments in this area and, most important, on their application to medical engineering. In vector optimization one considers optimization problems with a vectorvalued objective map and thus one has to compare elements in a linear space. If the linear space is the finite dimensional space Rm this can be done componentwise. That corresponds to the notion of an EdgeworthParetooptimal solution of a multiobjective optimization problem. Among the multitude of applications which can be modeled by such a multiobjective optimization problem, we present an application in intensity modulated radiation therapy and its solution by a numerical procedure. In case the linear space is arbitrary, maybe infinite dimensional, one may introduce a partial ordering which defines how elements are compared. Such problems arise for instance in magnetic resonance tomography where the number of Hermitian matrices which have to be considered for a control of the maximum local specific absorption rate can be reduced by applying procedures from vector optimization. In addition to a short introduction and the application problem, we present a numerical solution method for solving such vector optimization problems. A partial ordering can be represented by a convex cone which describes the set of directions in which one assumes that the current values are deteriorated. If one assumes that this set may vary dependently on the actually considered element in the linear space, one may replace the partial ordering by a variable ordering structure. This was for instance done in an application in medical image registration. We present a possibility of how to model such variable ordering structures mathematically and how optimality can be defined in such a case. We also give a numerical solution method for the case of a finite set of alternatives.
Vector Optimization with a Variable Ordering Structure
, 2009
"... In this work vector optimization problems are examined which have a variable ordering structure defined by a setvalued map which associates to each element in the objective space a cone of preferred or dominated directions. These considerations are motivated by a recent application in medical image ..."
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Cited by 1 (1 self)
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In this work vector optimization problems are examined which have a variable ordering structure defined by a setvalued map which associates to each element in the objective space a cone of preferred or dominated directions. These considerations are motivated by a recent application in medical image registration where the preferences vary dependently on the actually considered element in the objective space. Several optimality concepts and characterizations of optimal solutions are discussed. Existence and scalarization results as well as optimality conditions and duality results are presented.
MEDACO: Solving Multiobjective Combinatorial Optimization with Evolution, Decomposition and Ant Colonies.
"... We propose a novel multiobjective evolutionary algorithm, MEDACO, a shorter acronym for MOEA/DACO, combining ant colony optimization (ACO) and multiobjective evolutionary algorithm based on decomposition (MOEA/D). The motivation is to use the onlinelearning capabilities of ACO, according to the Re ..."
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We propose a novel multiobjective evolutionary algorithm, MEDACO, a shorter acronym for MOEA/DACO, combining ant colony optimization (ACO) and multiobjective evolutionary algorithm based on decomposition (MOEA/D). The motivation is to use the onlinelearning capabilities of ACO, according to the Reactive Search Optimization (RSO) paradigm of ”learning while optimizing”, to further improve the effectiveness of the original MOEA/D algorithms. Following other MOEA/Dlike algorithms, MEDACO decomposes a multiobjective optimization problem into a number of singleobjective optimization tasks solved by different iterated greedy construction processes (a.k.a. ants). Each ant has an individual heuristic information matrix and several neighboring ants, characterized by a similar combination of the individual objectives. All ants are divided into groups, with each group maintaining a different pheromone matrix. During the search, each ant records the best solution found so far for its subproblem. To construct a new solution, an ant combines information from its group’s pheromone matrix, its own heuristic information matrix and its current solution. Extensive experimental comparisons are executed. On the multiobjective 01 knapsack problem, MEDACO outperforms MOEA/DGA on all the nine test instances. Furthermore, we demonstrate that the heuristic information matrices in MEDACO are crucial to significantly improve the performance. On the biobjective traveling salesman problem, MEDACO performs much better than the previously proposed BicriterionAnt algorithm on the 12 test instances. We also critically evaluate the effects of the group, the neighborhood and the location information of current solutions on the performance of MEDACO.