Numerous existing deep learning models for time-series classification (TSC) tend to overlook the intricate interplay between higher-and lower-level semantic information. While the focus is often on extracting higher-level semantics from lower-level sources, the reciprocal influence of lower-level information on higher levels is undervalued. To address this, we propose an ensemble transitive bidirectional decoupled self-distillation (ETBiDecSD) method for TSC. ETBiDecSD enhances the robustness of higher-level semantic information using an average feature ensemble (AFE) method to amalgamate the output from each level. Simultaneously, the integrated features are transmitted to each lower level through a directional decoupled distillation (DD) structure. Additionally, to promote deep interaction between higher-and lower-level semantic information, ETBiDecSD introduces a transitive bidirectional DD (TBDD) structure, facilitating the transfer of target-class and nontarget-class knowledge between higher and lower levels. Experimental results demonstrate that whether a fully convolutional network (FCN) with four convolutional blocks or InceptionTime with four Inception blocks is used as the baseline, ETBiDecSD outperforms a quantity of well-established self-distillation algorithms across 85 widely used UCR2018 datasets, as evidenced by the metrics “win”/“tie”/“lose” and avg. rank, which are derived from accuracy andF1-scores. Notably, when compared to a nonself-distillation FCN, ETBiDecSD achieves “win”/“tie”/“lose” results of 64/4/17 in terms of accuracy and 65/4/16 in terms ofF1-score. Similarly, in comparison to a nonself-distillation InceptionTime, ETBiDecSD attains “win”/“tie”/“lose” results of 60/12/13 for accuracy and 57/12/16 forF1-score.

Multi-objective Bayesian optimization (MOBO) aims to optimize multiple competing objective functions in the expensive-to-evaluate scenario. The Expected Hypervolume Improvement (EHVI) is a commonly used acquisition function for MOBO and shows a good performance. However, the computation of EHVI becomes challenging as the number of objective functions grows. In this paper, we revisit the formulation of EHVI, as well as its multi-point counterpart qEHVI, and derive much simpler analytic expressions for them. The main contributions of this paper include: (1) first formulating EHVI as a particular hypervolume improvement, and thus immediately obtaining a formal proof of its NP-hardness, faster algorithms in both theory and practice, and more results on its derivatives; (2) first obtaining the analytic expressions of qEHVI for any q > 1 and m ≥ 2 where m is the number of objectives; and (3) demonstrating the advantages of our formulation over existing exact and approximation methods for computing EHVI and qEHVI through a large number of numerical experiments.

Deep Reinforcement Learning (DRL) has gained significant attention for its ability to solve combinatorial optimization problems, including the Traveling Salesman Problem (TSP). While traditional approaches assume known and deterministic cost coefficients, real-world scenarios often involve uncertainty, such as traffic conditions or varying travel times. To address this, we introduce a Multiobjective Robust Combinatorial Optimization (MO-RCO) model that incorporates robustness as an additional objective. By transforming a classical robust combinatorial optimization (RCO) problem into a multiobjective framework, MO-RCO allows decision-makers to explore optimal solutions via a Pareto front, while utilizing pre-trained multiobjective models to estimate robust solutions for new uncertain instances. This enables our method to balance optimality and robustness and provide a scalable solution for new RCO problems. Our results demonstrate the efficiency of MO-RCO in solving the min-max TSP with budget uncertainty, showcasing its potential for real-world applications in robust decision-making.

Over the years, various sophisticated deep learning algorithms have surfaced for multivariate time series classification (MTSC), notably the dual-network-based model. This model comprises two parallel networks tailored to time series data: one for local feature extraction and the other for global relation extraction. However, effectively integrating these dual networks poses a significant challenge. To address this, we propose a knowledge aggregation transformer network (KATN) for MTSC. KATN, composed of four aggregation transformer blocks, extracts abundant regularizations and connections hidden within the data. Each block incorporates a modified residual network (MResNet) for local feature extraction and a multi-head attention network for global relation extraction. Initially, the block merges MResNet's output feature with that of the multi-head attention network through an additive operation. Subsequently, it aligns features with a fully connected (i.e., dense) layer and activates neural units using the Gaussian error linear unit function. This strategic feature aggregation allows for capturing long-range dependencies among multiple variables in multivariate time series data. Experimental results demonstrate that KATN significantly outperforms 6 state-of-the-art transformer variants, achieving a ‘win’/‘tie’/‘lose’ record of 9/6/15 and securing the lowest AVG_rank score. Furthermore, when evaluated against 18 existing MTSC algorithms across 13 UEA datasets, KATN consistently delivers superior performance, attaining the lowest AVG_rank score among all compared methods.

In the realm of evolutionary multiobjective optimization, the hypervolume indicator serves as a crucial metric for assessing the quality of solution sets. Due to the high costs in hypervolume computation, hypervolume-based optimization algorithms always meet the challenge of finding a certain number of points in a given point set to maximize the hypervolume indicator, especially when there are many objectives. In response, the greedy decremental algorithm for hypervolume subset selection problem (gHSSD) has emerged as a noteworthy alternative. This paper introduces a general algorithm for gHSSD, applicable in any dimensionality above two. The proposed algorithm leverages a space partition tree and incorporates a once-build-multiple-use strategy, effectively reducing time complexity. We prove that the proposed algorithm has a time complexity of O((n-k+n)nd-12logn) where n is the number of points, k is the number of points to be reserved, and d the dimensionality. Theoretically, this complexity is competitive with the current best algorithms for d=3,4 and better than them for all 5≤d≤7. To validate our algorithm, we have conducted extensive tests on various random point sets and multiobjective optimization benchmarks. Experimental results suggest that our implementation is more efficient than or competitive with state-of-the-art algorithms on many instances as n increases for d=3,4.

Recently, numerous deep learning algorithms have addressed wearable human activity recognition (HAR), but they often struggle with efficient knowledge transfer to lightweight models for mobile devices. Knowledge distillation (KD) is a popular technique for model compression, transferring knowledge from a complex teacher to a compact student. Most existing KD algorithms consider homogeneous architectures, hindering performance in heterogeneous setups. This is an under-explored area in wearable HAR. To bridge this gap, we propose a heterogeneous mutual KD (HMKD) framework for wearable HAR. HMKD establishes mutual learning within the intermediate and output layers of both teacher and student models. To accommodate substantial structural differences between teacher and student, we employ a weighted ensemble feature approach to merge the features from their intermediate layers, enhancing knowledge exchange within them. Experimental results on the HAPT, WISDM, and UCI_HAR datasets show HMKD outperforms ten state-of-the-art KD algorithms in terms of classification accuracy. Notably, with ResNetLSTMaN as the teacher and MLP as the student, HMKD increases by 9.19% in MLP's $F_{1}$ score on the HAPT dataset.

Many multiobjective evolutionary algorithms (MOEAs) have been proposed for dealing with various problem difficulties in multiobjective optimization over the past three decades. However, none of them can perform best for all problem difficulties. When solving a certain multiobjective optimization problem (MOP), a good multiobjective optimizer should take its problem features into account. When the problem features are unknown in advance, it is difficult to choose an appropriate algorithm as the prior solver. In this paper, we propose a Q-learning evolutionary multiobjective framework, denoted by QL-MOEA, to solve the MOPs with both separable variables and interacting variables. In QL-MOEA, either NSGA-II or MOEA/D is adaptively selected by intelligent agent in different stages of the evolution of population. Our experimental results show that QL-MOEA outperforms the baseline NSGA-II or MOEA/D in convergence speed.

The hypervolume contribution is widely used in indicator-based multiobjective algorithms. We propose an algorithm to compute exact 4-D hypervolume contributions for a set of n points in O(n32logn) time. Our algorithm improves the currently best time complexity O(n2) by O(nlogn), and it is the first algorithm of subquadratic time for this problem. Our algorithm is built upon a space partition method in computational geometry and a geometric structure called the anchored gradient. We also propose a new space partition strategy to reduce the practical running time and the space overhead of this algorithm. Experimental results on a variety of test instances show that our proposed algorithm performs better than the existing state-of-the-art algorithm especially on point sets with cliff or other irregular properties.

The research on expensive multiobjective optimization has attracted particular attention in the area of multiobjective evolutionary computation. Many existing multiobjective evolutionary algorithms (MOEAs) are only suited for small-scale expensive multiobjective optimization problems (MOPs) with less than ten decision variables. The main reason lies in the fact that some optimization techniques used in expensive MOEAs, such as Gaussian Process (GP), are not applicable for exploring high-dimensional search space. The naive way to overcome this difficulty is to convert a high-dimensional expensive MOP into a low-dimensional MOP, which can be solved by existing expensive MOEAs efficiently. In this paper, we investigate the combination of MOEA/D with a weighted optimization framework (WOF) and GP, denoted by MOEA/D-WOFGP, for solving high-dimensional expensive MOPs, where the WOF converts a high-dimensional MOP into a low-dimensional search space of weight variables, and the GP-based learning method is used to predict high-quality solutions within a limited number of function evaluations. Some experiments are conducted to compare the performance of MOEA/D-WOFGP with other expensive MOEAs assisted by variable grouping. Our experimental results show that MOEA/D-WOFGP is advantageous when dealing with high-dimensional expensive MOPs.

Approximating the Pareto fronts of large-scale multiobjective optimization problems (LSMOPs) is a very changeling task due to their huge search spaces caused by the large number of decision variables. It is a commonly-used idea that large scale optimization problems are often transformed into small scale optimization problems that can be solved by existing optimization methods. In this paper, we investigate an improved version of MOEA/D with dimensionality reduction for large-scale multiobjective optimization, denoted by LS-MOEA/D-GSD. The major ideas in our proposed method focus on two aspects. On the one hand, the original search space of LSMOPs is transformed into a small-scale MOP on weight variables of several guided search directions via the genetic operators in differential evolution. On the other hand, the computational resources are allocated to both the original search space and reduced search space. Some experiments are conducted to test the performance of our proposed algorithm on the well-known large scale multiobjective test suites, i.e., LSMOP1-9 with up to 1000 variables. Our experimental results show that our algorithm outperforms several state-of-the-art multiobjective evolutionary algorithms for large scale multiobjective optimization.

Multiobjective evolutionary algorithms (MOEAs) have progressed significantly in recent decades, but most of them are designed to solve unconstrained multiobjective optimization problems. In fact, many real-world multiobjective problems contain a number of constraints. To promote research on constrained multiobjective optimization, we first propose a problem classification scheme with three primary types of difficulty, which reflect various types of challenges presented by real-world optimization problems, in order to characterize the constraint functions in constrained multiobjective optimization problems (CMOPs). These are feasibility-hardness, convergence-hardness, and diversity-hardness. We then develop a general toolkit to construct difficulty adjustable and scalable CMOPs (DAS-CMOPs, or DAS-CMaOPs when the number of objectives is greater than three) with three types of parameterized constraint functions developed to capture the three proposed types of difficulty. In fact, the combination of the three primary constraint functions with different parameters allows the construction of a large variety of CMOPs, with difficulty that can be defined by a triplet, with each of its parameters specifying the level of one of the types of primary difficulty. Furthermore, the number of objectives in this toolkit can be scaled beyond three. Based on this toolkit, we suggest nine difficulty adjustable and scalable CMOPs and nine CMaOPs, to be called DAS-CMOP1-9 and DAS-CMaOP1-9, respectively. To evaluate the proposed test problems, two popular CMOEAs-MOEA/D-CDP (MOEA/D with constraint dominance principle) and NSGA-II-CDP (NSGA-II with constraint dominance principle) and two popular constrained many-objective evolutionary algorithms (CMaOEAs)-C-MOEA/DD and C-NSGA-III-are used to compare performance on DAS-CMOP1-9 and DAS-CMaOP1-9 with a variety of difficulty triplets, respectively. The experimental results reveal that mechanisms in MOEA/D-CDP may be more effective in solving convergence-hard DAS-CMOPs, while mechanisms of NSGA-II-CDP may be more effective in solving DAS-CMOPs with simultaneous diversity-, feasibility-, and convergence-hardness. Mechanisms in C-NSGA-III may be more effective in solving feasibility-hard CMaOPs, while mechanisms of C-MOEA/DD may be more effective in solving CMaOPs with convergence-hardness. In addition, none of them can solve these problems efficiently, which stimulates us to continue to develop new CMOEAs and CMaOEAs to solve the suggested DAS-CMOPs and DAS-CMaOPs.

Decomposition-based multi-objective evolutionary algorithm has been acknowledged as a promising paradigm for multi-objective optimization problems. Nevertheless, its performance deteriorates seriously when the number of objectives increases. To improve its performance, generating high-quality solution is vital. Acknowledging the success of hybridizing different recombination operators, a selection strategy to choose from a set of differential evolution (DE) operators is adopted in this paper. The selection strategy could combine the advantages of these DE operators. Yet, the performance of DE operators depends highly on their control parameters, which should be tuned adaptively along the search process to fully explore their search abilities. An adaptive parameter tuning strategy is hereby proposed by estimating a Cauchy and a normal distribution from history information for the control parameters, respectively. Experimental comparison using DTLZ1-DTLZ4, with the number of objectives ranging from three to ten, is carried out between six state-of-the-art algorithms and the developed algorithm. Empirical results justify the outperformance of the developed algorithm against the compared algorithms in terms of some commonly-used performance metrics.

Optimization problems with variable-length decision space are a class of challenging optimization problems derived from some real-world applications, such as the composite laminate stacking problem and the sensor coverage problem. Unlike other optimization problems, the solutions in these problems might be represented as the vectors with different variable size (i.e., dimensionality). So far, some research efforts have been done on the use of evolutionary algorithms (EAs) for solving single objective variable-length optimization problems. In fact, the variable-length problem difficulty can also exist in multiobjective optimization. However, such challenging problems have not yet gained much attention in the area of evolutionary multiobjective optimization. To facilitate the research on the variable-length Pareto optimization, we first suggest a systematic toolkit for constructing benchmark multiobjective test problems with variable-length feature in this paper. Then, we also propose a variable-length multiobjective EA based on a two-level decomposition strategy, which decomposes a multiobjective optimization problem in terms of the penalty boundary intersection search directions and the dimensionality of variables. The performance of our proposed algorithm and the other three state-of-the-art algorithms on these problems are compared. To further show the effectiveness of our proposed algorithm, some experimental results on a bi-objective laminate stacking optimization problem are also reported and analyzed.

This paper introduces a differential prediction model to predict the varying Pareto-Optimal Solutions (POS) when solving dynamic multiobjective optimization problems (DMOPs). In dynamic multiobjective optimization problems, several competing objective functions and/or constraints change over time. As a consequence, the Pareto-Optimal Solutions and/or Pareto-Optimal Front may vary over time. The differential prediction model is used to forecast the shift vector in the decision space of the centroid in the population through the centroid's historical locations in three previous environments. This differential prediction model is incorporated into a multiobjective evolutionary algorithm based on decomposition to solve DMOPs. After detecting the environmental change, half of individuals in the population are forecasted their new positions in the decision space by using the differential prediction model and the others' positions are retained. The proposed model is tested on a number of typical benchmark problems with several dynamic characteristics. Experimental results show that the proposed model is competitively in comparisons with the other state-of-the-art models or approaches that were proposed for solving DMOPs.

Iterative thresholding is a dominating strategy for sparse optimization problems. The main goal of iterative thresholding methods is to find a so-called -sparse solution. However, the setting of regularization parameters or the estimation of the true sparsity are nontrivial in iterative thresholding methods. To overcome this shortcoming, we propose a preference-based multiobjective evolutionary approach to solve sparse optimization problems in compressive sensing. Our basic strategy is to search the knee part of weakly Pareto front with preference on the true -sparse solution. In the noiseless case, it is easy to locate the exact position of the -sparse solution from the distribution of the solutions found by our proposed method. Therefore, our method has the ability to detect the true sparsity. Moreover, any iterative thresholding methods can be used as a local optimizer in our proposed method, and no prior estimation of sparsity is required. The proposed method can also be extended to solve sparse optimization problems with noise. Extensive experiments have been conducted to study its performance on artificial signals and magnetic resonance imaging signals. Our experimental results have shown that our proposed method is very effective for detecting sparsity and can improve the reconstruction ability of existing iterative thresholding methods.

In recent years, research interests have been paid in solving real-world optimization problems with variable-length representation. For population-based optimization algorithms, the challenge lies in maintaining diversity in sizes of solutions and in designing a suitable recombination operator for achieving an adequate diversity. In dealing with multiple conflicting objectives associated with a variable-length problem, the resulting multiple trade-off Pareto-optimal solutions may inherently have different variable sizes. In such a scenario, the fixed recombination and mutation operators may not be able to maintain large-sized solutions, thereby not finding the entire Pareto-optimal set. In this paper, we first construct multiobjective test problems with variable-length structures, and then analyze the difficulties of the constructed test problems by comparing the performance of three state-of-the-art multiobjective evolutionary algorithms. Our preliminary experimental results show that MOEA/D-M2M shows good potential in solving the multiobjective test problems with variable-length structures due to its diversity strategy along different search directions. Our correlation analysis on the Pareto solutions with variable sizes in the Pareto front indicates that mating restriction is necessary in solving variable-length problem.

The bias feature is a major factor that makes a multiobjective optimization problem (MOP) difficult for multiobjective evolutionary algorithms (MOEAs). To deal with this problem feature, an algorithm should carefully balance between exploration and exploitation. The decomposition-based MOEA decomposes an MOP into a number of single objective subproblems and solves them in a collaborative manner. Single objective optimizers can be easily used in this algorithm framework. Covariance matrix adaptation evolution strategy (CMA-ES) has proven to be able to strike good balance between the exploration and the exploitation of search space. This paper proposes a scheme to use both differential evolution (DE) and covariance matrix adaptation in the MOEA based on decomposition. In this scheme, single objective optimization problems are clustered into several groups. To reduce the computational overhead, only one subproblem from each group is selected to optimize by CMA-ES while other subproblems are optimized by DE. When an evolution strategy procedure meets some stopping criteria, it will be reinitialized and used for solving another subproblem in the same group. A set of new multiobjective test problems with bias features are constructed in this paper. Extensive experimental studies show that our proposed algorithm is suitable for dealing with problems with biases.

This paper proposes a new constraint handling method named Angle-based Constrained Dominance Principle (ACDP). Unlike the original Constrained Dominance Principle (CDP), this approach adopts the angle information of the objective functions to enhance the population's diversity in the infeasible region. To be more specific, given two infeasible solutions, if the angle of the solutions is greater than a given threshold, they are considered to be non-dominated by each other. For a feasible solution and an infeasible solution, if the angle of the solutions is less than a given threshold, the feasible solution is better, otherwise they are non-dominated. To verify the proposed constraint handling approach ACDP, eight test problems CMOP1 to CMOP8 are introduced. The suggested algorithm MOEA/D-ACDP is compared with MOEA/D-CDP and NSGA-II-CDP on CMOP1 to CMOP8. The experimental results demonstrate that ACDP performs better than CDP in the framework of MOEA/D, and MOEA/D-ACDP is significantly better than NSGA-II-CDP, especially on the test instances with the very low ratio of feasible region against the whole objective space.

Solving sparse optimization problems via regularization frameworks is the dominant methodology for reconstructing sparse signals in the area of compressive sensing. In recent a few years, the use of multiobjective evolutionary algorithms (MOEAs) for sparse optimization has also attracted some research interests. Under the multiobjective framework, the loss term (error) and the regularization term (sparsity) are treated as two separate objective functions. So far, two popular multiobjective frameworks, NSGA-II and MOEA/D, have been used for sparse optimization. In this paper, we further develop a new MOEA/D variant for sparse reconstruction and sparsity detection, which involves three phases - approximating Pareto front (PF) in a chain order (phase 1) and in a random order (phase 2), and exploiting a knee region (phase 3 - optional). Our experimental results show that our proposed method is more effective than the earlier version of MOEA/D and the HALF solver in sparse signal reconstruction and sparsity detection.

MOEA/D is a decomposition-based multiobjective evolutionary algorithm that has attracted much attention in recent years. Its performance depends on the setting of weight vectors which are used for defining subproblems. In the case of irregular Pareto fronts (e.g, disconnected or degenerated), fixed setting of weight vectors in MOEA/D may not work well. In this paper, we propose an improved MOEA/D with both random and fixed weight vectors. Moreover, an external archive based on a modified ε-dominance strategy is used for storing nondominated solutions found by the proposed algorithm and assisting the generation of random weight vectors. Some experiments have been conducted to verify the efficiency and effectiveness of the improved MOEA/D on benchmark multiobjective test problems with irregular Pareto fronts. The experimental results show that the overall performance of the proposed algorithm is better than baseline MOEA/D and NSGA-II.

This paper studies how to use two reproduction operators with different characteristics for balancing the convergence and the diversity in MOEA/D. We consider two operators. One is a differential evolution and polynomial mutation, and the other is a neighbor learning and inversion mutation. We show that these two operators have different search abilities. Then we propose a scheme to use these two operators in our recently proposed MOEA/D-GR framework. We test the proposed algorithm on some benchmark problems to demonstrate its effectiveness.

It is well-established that the shapes of Pareto-optimal fronts (POFs) can affect the performance of some multiobjective optimization methods. The most well-known characteristics on the shape of POFs are convexity and discontinuity. In this paper, we investigate the construction of multiobjective test problems with complicated POFs, of which its local parts could have mixed dimensionalities. For example, in the case of 3 objectives, some parts of POFs can be 1-D curves while others could be 2-D surfaces. We formulate eight test problems, called CPFT1-8, with such a feature. To study the difficulties of these test problems, we conducted some experiments with two state-of-the-art algorithms MOEA/D and NSGA-II, and analyzed their performances.

A multi-objective optimization problem can be solved by decomposing it into one or more single objective subproblems in some multi-objective metaheuristic algorithms. Each subproblem corresponds to one weighted aggregation function. For example, MOEA/D is an evolutionary multi-objective optimization (EMO) algorithm that attempts to optimize multiple subproblems simultaneously by evolving a population of solutions. However, the performance of MOEA/D highly depends on the initial setting and diversity of the weight vectors. In this paper, we present an improved version of MOEA/D, called EMOSA, which incorporates an advanced local search technique (simulated annealing) and adapts the search directions (weight vectors) corresponding to various subproblems. In EMOSA, the weight vector of each subproblem is adaptively modified at the lowest temperature in order to diversify the search toward the unexplored parts of the Pareto-optimal front. Our computational results show that EMOSA outperforms six other well established multi-objective metaheuristic algorithms on both the (constrained) multi-objective knapsack problem and the (unconstrained) multi-objective traveling salesman problem. Moreover, the effects of the main algorithmic components and parameter sensitivities on the search performance of EMOSA are experimentally investigated.

Recently, the research on quantum-inspired evolutionary algorithms (QEA) has attracted some attention in the area of evolutionary computation. QEA use a probabilistic representation, called Q-bit, to encode individuals in population. Unlike standard evolutionary algorithms, each Q-bit individual is a probability model, which can represent multiple solutions. Since probability models store global statistical information of good solutions found previously in the search, QEA have good potential to deal with hard optimization problems with many local optimal solutions. So far, not much work has been done on evolutionary multi-objective (EMO) algorithms with probabilistic representation. In this paper, we investigate the performance of two state-of-the-art EMO algorithms - MOEA/D and NSGA-II, with probabilistic representation based on pheromone trails, on the multi-objective travelling salesman problem. Our experimental results show that MOEA/D and NSGA-II with probabilistic presentation are very promising in sampling high-quality offspring solutions and in diversifying the search along the Pareto fronts.

MOEA/D is a generic multiobjective evolutionary optimization algorithm. MOEA/D needs a approach to decompose a multiobjective optimization problem into a number of single objective optimization problems. The commonly-used weighted sum approach and the Tchebycheff approach may not be able to handle disparately scaled objectives. This paper suggests a new decomposition approach, called NBI-style Tchebycheff approach, for MOEA/D to deal with such objectives. A portfolio management MOP has been used as an example to test the effectiveness of MOEA/D with NBI-style Tchebycheff approach.

This paper describes the idea of MOEA/D and proposes a strategy for allocating the computational resource to different subproblems in MOEA/D. The new version of MOEA/D has been tested on all the CEC09 unconstrained MOP test instances.

Partly due to lack of test problems, the impact of the Pareto set (PS) shapes on the performance of evolutionary algorithms has not yet attracted much attention. This paper introduces a general class of continuous multiobjective optimization test instances with arbitrary prescribed PS shapes, which could be used for studying the ability of multiobjective evolutionary algorithms for dealing with complicated PS shapes. It also proposes a new version of MOEA/D based on differential evolution (DE), i.e., MOEA/D-DE, and compares the proposed algorithm with NSGA-II with the same reproduction operators on the test instances introduced in this paper. The experimental results indicate that MOEA/D could significantly outperform NSGA-II on these test instances. It suggests that decomposition based multiobjective evolutionary algorithms are very promising in dealing with complicated PS shapes.

In this paper, we propose a population-based implementation of simulated annealing to tackle multi-objective optimisation problems, in particular those of combinatorial nature. The proposed algorithm is called Evolutionary Multi-objective Simulated Annealing Algorithm (EMOSA), which combines local and evolutionary search by incorporating two distinctive features. The first feature is to tune the weight vectors of scalarizing functions (i.e., search directions) for selection during local search using a two-phase strategy. The second feature is the competition between members of the current population with similar weight vectors. We compare the proposed algorithm to three other multi-objective simulated annealing algorithms and also to the Pareto archived evolutionary strategy (PAES). Experiments are carried out on a set of bi-objective travelling salesman problem (TSP) instances with convex or nonconvex Pareto-optimal fronts. Our experimental results demonstrate that the two-phase tuning of weight vectors and the competition between individuals make a significant contribution to the improved performance of EMOSA.

Decomposition is a basic strategy in traditional multiobjective optimization. However, it has not yet been widely used in multiobjective evolutionary optimization. This paper proposes a multiobjective evolutionary algorithm based on decomposition (MOEA/D). It decomposes a multiobjective optimization problem into a number of scalar optimization subproblems and optimizes them simultaneously. Each subproblem is optimized by only using information from its several neighboring subproblems, which makes MOEA/D have lower computational complexity at each generation than MOGLS and nondominated sorting genetic algorithm II (NSGA-II). Experimental results have demonstrated that MOEA/D with simple decomposition methods outperforms or performs similarly to MOGLS and NSGA-II on multiobjective 0-1 knapsack problems and continuous multiobjective optimization problems. It has been shown that MOEA/D using objective normalization can deal with disparately-scaled objectives, and MOEA/D with an advanced decomposition method can generate a set of very evenly distributed solutions for 3-objective test instances. The ability of MOEA/D with small population, the scalability and sensitivity of MOEA/D have also been experimentally investigated in this paper.