Multiobjective evolutionary algorithms based on decomposition (MOEA/D) decompose a multiobjective optimization problem into a set of simple optimization subproblems and solve them in a collaborative manner. A replacement scheme, which assigns a new solution to a subproblem, plays a key role in balancing diversity and convergence in MOEA/D. This paper proposes a global replacement scheme which assigns a new solution to its most suitable subproblems. We demonstrate that the replacement neighborhood size is critical for population diversity and convergence, and develop an approach for adjusting this size dynamically. A steady-state algorithm and a generational one with this approach have been designed and experimentally studied. The experimental results on a number of test problems have shown that the proposed algorithms have some advantages.

Real-world applications typically have multiple sparse reconstruction tasks to be optimized. In order to exploit the similar sparsity pattern between different tasks, this paper establishes an evolutionary multitasking framework to simultaneously optimize multiple sparse reconstruction tasks using a single population. In the proposed method, the evolutionary algorithm aims to search the locations of nonzero components or rows instead of searching sparse vector or matrix directly. Then the within-task and between-task genetic transfer operators are employed to reinforce the exchange of genetic material belonging to the same or different tasks. The proposed method can solve multiple measurement vector problems efficiently because the length of decision vector is independent of the number of measurement vectors. Finally, a case study on hyperspectral image unmixing is investigated in an evolutionary multitasking setting. It is natural to consider a sparse unmixing problem in a homogeneous region as a task. Experiments on signal reconstruction and hyperspectral image unmixing demonstrate the effectiveness of the proposed multitasking framework for sparse reconstruction.

This paper proposes a Gaussian process (GP) based co-sub-Pareto front surrogate augmentation strategy for evolutionary optimization of computationally expensive multiobjective problems. In the proposed algorithm, a multiobjective problem is decomposed into a number of subproblems, the solution of each of which is used to approximate a portion or sector of the Pareto front (i.e., a subPF). Thereafter, a multitask GP model is incorporated to exploit the correlations across the subproblems via joint surrogate model learning. A novel criterion for the utility function is defined on the surrogate landscape to determine the next candidate solution for evaluation using the actual expensive objectives. In addition, a new management strategy for the evaluated solutions is presented for model building. The novel feature of our approach is that it infers multiple subproblems jointly by exploiting the possible dependencies between them, such that knowledge can be transferred across subPFs approximated by the subproblems. Experimental studies under several scenarios indicate that the proposed algorithm outperforms state-of-the-art multiobjective evolutionary algorithms for expensive problems. The parameter sensitivity and effectiveness of the proposed algorithm are analyzed in detail.

The DTLZ1-DTLZ4 problems are by far one of the most commonly used test problems in the validation and comparison of multiobjective optimization evolutionary algorithms. However, very recently, it has been pointed out that they have the following two special characteristics: the regularly-oriented Pareto front shape and the single distance function. As a modification of them, this paper presents a new set of test problems mDTLZ1-mDTLZ4 to avoid the two special characteristics. Using these new test problems, we investigate the performance of three representative multiobjective evolutionary algorithms NSGA-II, MOEA/D-Tch and SMS-EMOA. Experimental results indicate that the performance of NSGA-II and MOEA/D-Tch deteriorates on mDTLZ1-mDTLZ4. Subsequently, our analysis reveals that there exist the hardly-dominated boundaries in each of mDTLZ1-mDTLZ4, which hinder the approximation of Pareto dominance based algorithms and Tchebycheff-decomposition based algorithms. Furthermore, we summarize that the hardly-dominated boundary should be an often encountered problem feature in multiobjective optimization. Last but not least, we point out and validate some coping strategies for dominance based algorithms and decomposition based algorithms to overcome the challenges caused by the hardly-dominated boundary.

Algorithm design is crucial for effective problem-solving across various domains. The advent of Large Language Models (LLMs) has notably enhanced the automation and innovation within this field, offering new perspectives and promising solutions. In just a few years, this integration has yielded remarkable progress in areas ranging from combinatorial optimization to scientific discovery. Despite this rapid expansion, a holistic understanding of the field is hindered by the lack of a systematic review, as existing surveys either remain limited to narrow sub-fields or with different objectives. This article seeks to provide a systematic review of algorithm design with LLMs. We introduce a taxonomy that categorizes the roles of LLMs as optimizers, predictors, extractors, and designers, analyzing the progress, advantages, and limitations within each category. We further synthesize literature across the three phases of the algorithm design pipeline and across diverse algorithmic applications that define the current landscape. Finally, we outline key open challenges and opportunities to guide future research.