Multi-objective optimization problems (MOPs) are prevalent in numerous real-world applications. Recently, Pareto Set Learning (PSL) has emerged as a powerful paradigm for solving MOPs. PSL can produce a neural network for modeling the set of all Pareto optimal solutions. However, applying PSL to black-box objectives, particularly those exhibiting non-separability, high dimensionality, and/or other complex properties, remains very challenging. To address this issue, we propose leveraging evolution strategies (ESs), a class of specialized black-box optimization algorithms, within the PSL paradigm. Traditional ESs capture the complex dimensional dependencies less efficiently, which can significantly hinder their performance in PSL. To tackle this issue, we suggest encapsulating the dependencies within a neural network, which is then trained using a novel gradient estimation method. The proposed method, termed Neural-ES, is evaluated using a bespoke benchmark suite for black-box PSL. Experimental comparisons with other methods demonstrate the efficiency of Neural-ES, underscoring its ability to learn the Pareto sets of challenging black-box MOPs.

The multi/many-objective traveling salesman problem (MOTSP), which is NP-hard, can be found in many real-world applications. The Lin–Kernighan (LK) algorithm, as one of the most successful local search (LS) methods for the single-objective traveling salesman problem, adopts a variable neighborhood LS. However, LK cannot be directly applied to the decomposition-based multiobjective optimization framework due to its incapability of effective knowledge transfer among different subproblems, especially for problems with more than two objectives. In this article, we propose an algorithm, called decomposition-based multiobjective LK heuristic with neighborhood structure transfer (NST-MOLK) for MOTSP. In NST-MOLK, the knowledge of a neighborhood structure has been transferred to enhance the efficiency and effectiveness of LK. The experimental studies have been conducted on both benchmark and real-world instances constructed based on the flight prices of seven airlines and 266 airports of different cities in China. Experimental results show that NST-MOLK outperforms both classical and state-of-the-art algorithms significantly. It has also been verified that neighborhood structure transfer can effectively improve the performance of NST-MOLK.

Many practical multiobjective optimization problems have a nested bilevel structure in variables, which can be modeled as bilevel multiobjective optimization problems (BLMOPs). In this article, a cooperative coevolution (CC) with knowledge-based variable decomposition, called bilevel multiobjective CC (BLMOCC), is proposed for BLMOPs. In BLMOCC, the variable interactions are represented by an interaction matrix. The perturbation-based variable decomposition combined with the matrix completion approach has been designed for dynamically discovering the correlation among the bilevel variables, based on which the variables are divided into different groups. To further handle possible weak correlations among various groups of variables, a CC has been adopted for optimizing them in a collaborative way. In experimental studies, BLMOCC is compared with a nested method (NS) and a state-of-the-art algorithm (H-BLEMO) on a set of benchmark problems. The effects of each component in BLMOCC have also been verified by comparing it with its three variants. The experimental results demonstrate that BLMOCC has the best performance among all the compared algorithms. In addition, BLMOCC has also been applied to a real-world management decision-making problem, which further validates its efficiency and effectiveness.

How to evaluate Pareto front approximations generated by multi/many-objective optimizers is a critical issue in the field of multiobjective optimization. Currently, there exist two types of comprehensive quality indicators (i.e., volume-based and distance-based indicators). Distance-based indicators, such as inverted generational distance (IGD), are usually computed by summing up the distance of each reference point to its nearest solution. Their high computational efficiency leads to their prevalence in many-objective optimization. However, in the existing distance-based indicators, the distributions of the solution sets are usually neglected, leading to their lack of ability to well distinguish between different solution sets. This phenomenon may become even more severe in high-dimensional space. To address such an issue, a kernel-based indicator (KBI) is proposed as a comprehensive indicator. Different from other distance-based indicators, a kernel-based maximum mean discrepancy is adopted in KBI for directly measuring the difference that can characterize the convergence, spread, and uniformity of two sets, i.e., the solution set and reference set, by embedding them in reproducing kernel Hilbert space (RKHS). As a result, KBI not only reflects the distance between the solution set and the reference set but also can reflect the distribution of the solution set itself. In addition, to maintain the desirable weak Pareto compliance property of KBI, a nondominated set reconstruction approach is also proposed to shift the original solution set. The detailed theoretical and experimental analysis of KBI is provided in this article. The properties of KBI have also been analyzed by the optimal$\mu $-distribution.

Optimization modeling of real-world application problems usually involves noise from various sources. Noisy optimization imposes challenges to optimization methods since the objective values can be different for multiple evaluations. In this article, we propose a novel online population size learning (OPL) technique of evolution strategies for handling noisy optimization problems. By re-evaluating a fraction of the candidates, we measure the strength of noise level of the re-evaluated candidate solutions and adapt the population size according to the noise level. The proposed OPL combines the advantages of both explicit averaging by re-evaluations and the implicit averaging by large population size and overcomes their limitations. We incorporate it with the covariance matrix adaptation evolution strategy (CMA-ES) and obtain OPL-CMA-ES. Compared with the existing noise handling technique, the proposed OPL is much simpler in both concepts and computation. We conduct comprehensive experiments to evaluate the algorithm’s performance on standard problems with Gaussian noise. We further evaluate the performance of OPL-CMA-ES on the black-box optimization benchmarks (BBOBs) noisy testbed, which is a standard platform for comparing black-box optimization algorithms, compared with the state-of-the-art noise-handling algorithms. The experimental results show that OPL-CMA-ES achieves remarkable performance and outperforms the compared variants.

To improve deep neural net hyper-parameter optimization we develop a deterministic surrogate optimization algorithm as an efficient alternative to Bayesian optimization. A deterministic Radial Basis Function (RBF) surrogate model is built to interpolate previously evaluated points, and this surrogate model is incrementally updated in each iteration. The stochastic algorithm CMA-ES is used to search the acquisition function based on the surrogate. The acquisition function at a point is based on a weighted average of the surrogate at x and the minimum distance from x to a previously evaluated point. We evaluate the proposed algorithm RBF-CMA on hyper-parameter optimization tasks for deep convolutional neural networks on datasets of CIFAR-10, SVHN, and CIFAR-100. We show that RBF-CMA achieves a promising performance especially when the search space dimension is high in comparison to other algorithms including GP-EI, GP-LCB, and SMBO.

Covariance matrix adaptation evolution strategy (CMA-ES) is a successful gradient-free optimization algorithm. Yet, it can hardly scale to handle high-dimensional problems. In this paper, we propose a fast variant of CMA-ES (Fast CMA-ES) to handle large-scale black-box optimization problems. We approximate the covariance matrix by a low-rank matrix with a few vectors and use two of them to generate each new solution. The algorithm achieves linear internal complexity on the dimension of search space. We illustrate that the covariance matrix of the underlying distribution can be considered as an ensemble of simple models constructed by two vectors. We experimentally investigate the algorithm's behaviors and performances. It is more efficient than the CMA-ES in terms of running time. It outperforms or performs comparatively to the variant limited memory CMA-ES on large-scale problems. Finally, we evaluate the algorithm's performance with a restart strategy on the CEC'2010 large-scale global optimization benchmarks, and it shows remarkable performance and outperforms the large-scale variants of the CMA-ES.

In this paper, we propose a computationally efficient variant of elitist covariance matrix evolution strategy for continuous local search in high dimensional space. It focuses on searching in a low-dimensional subspace expanded by a small number of promising search directions. This leads to the linear internal computational complexity of each iteration, which enables the algorithm to scale to high dimensional problems. We conduct comprehensive experiments to evaluate the parameter sensitivity and the algorithm’s performance. The experimental results validate that the proposed algorithm reduces the running time by a factor of ten, and it can be easily scaled up to n>1000 on a set of commonly used test functions.

Multi-task learning is a powerful method for solving multiple correlated tasks simultaneously. However, it is often impossible to find one single solution to optimize all the tasks, since different tasks might conflict with each other. Recently, a novel method is proposed to find one single Pareto optimal solution with good trade-off among different tasks by casting multi-task learning as multiobjective optimization. In this paper, we generalize this idea and propose a novel Pareto multi-task learning algorithm (Pareto MTL) to find a set of well-distributed Pareto solutions which can represent different trade-offs among different tasks. The proposed algorithm first formulates a multi-task learning problem as a multiobjective optimization problem, and then decomposes the multiobjective optimization problem into a set of constrained subproblems with different trade-off preferences. By solving these subproblems in parallel, Pareto MTL can find a set of well-representative Pareto optimal solutions with different trade-off among all tasks. Practitioners can easily select their preferred solution from these Pareto solutions, or use different trade-off solutions for different situations. Experimental results confirm that the proposed algorithm can generate well-representative solutions and outperform some state-of-the-art algorithms on many multi-task learning applications.

We propose an evolution strategy algorithm using a sparse plus low rank model for large-scale optimization in this paper. We first develop a rank one evolution strategy using a single principal search direction. It is of linear complexity. Then we extend it to multiple search directions, and develop a rank-${m}$evolution strategy. We illustrate that the principal search direction accumulates the natural gradients with respect to the distribution mean, and acts as a momentum term. Further, we analyze the optimal low rank approximation to the covariance matrix, and experimentally show that the principal search direction can effectively learn the long valley of the function with predominant search direction. Then we investigate the effects of Hessian on the algorithm performance. We conduct experiments on a class of test problems and the CEC’2010 LSGO benchmarks. The experimental results validate the effectiveness of our proposed algorithms.

Covariance matrix adaptation evolution strategies (CMA-ES) is a powerful optimizer. In this paper, we propose an efficient rank-1 update for the Cholesky covariance matrix adaptation evolution strategy (Cholesky CMA-ES) using an auxiliary evolution path. It accumulates the average mutation vector corresponding to the current search direction, which is used to update the evolution path. It is used to update the Cholesky factor. It avoids to maintain the additional inverse Cholesky factor, and reduces the computational complexity in the update procedure to a half. Further, we experimentally show that the auxiliary evolution path approximates to the inverse vector of the evolution path in terms of inverse Cholesky factor in the procedure. We experimentally show that the proposed method achieves comparative or even better performances on the test problems.