Neural Information Processing Systems http://nips.cc/

Understanding the relationships between objectives in a multiobjective optimisation problem is important for developing tailored and efficient solving techniques. In particular, when tackling combinatorial optimisation problems with many objectives, that arise in real-world logistic scenarios, better support for the decision maker can be achieved through better understanding of the often complex fitness landscape. This paper makes a contribution in this direction by presenting a technique that allows a visualisation and analysis of the local and global relationships between objectives in optimisation problems with many objectives. The proposed technique uses four steps: First, the global pairwise relationships are analysed using the Kendall correlation method; then, the ranges of the values found on the given Pareto front are estimated and assessed; next, these ranges are used to plot a map using Gray code, similar to Karnaugh maps, that has the ability to highlight the trade-offs between multiple objectives; and finally, local relationships are identified using scatter plots. Experiments are presented for three combinatorial optimisation problems: multiobjective multidimensional knapsack problem, multiobjective nurse scheduling problem, and multiobjective vehicle routing problem with time windows . Results show that the proposed technique helps in the gaining of insights into the problem difficulty arising from the relationships between objectives.

These preferences are defined based on the stability of the obtained solutions with respect to preferred objective functions.

We then compute the average NHV across the instances used for testing.

We present a novel approach to help decision-makers efficiently identify preferred solutions from the Pareto set of a multi-objective optimization problem. Our method uses a Bayesian model to estimate the decision-maker's utility function based on pairwise comparisons. Aided by this model, a principled elicitation strategy selects queries interactively to balance exploration and exploitation, guiding the discovery of high-utility solutions. The approach is flexible: it can be used interactively or a posteriori after estimating the Pareto front through standard multi-objective optimization techniques. Additionally, at the end of the elicitation phase, it generates a reduced menu of high-quality solutions, simplifying the decision-making process. Through experiments on test problems with up to nine objectives, our method demonstrates superior performance in finding high-utility solutions with a small number of queries. We also provide an open-source implementation of our method to support its adoption by the broader community.

Multi-objective optimization involving Quadratic Unconstrained Binary Optimization (QUBO) problems arises in various domains. A fundamental challenge in this context is the effective balancing of multiple objectives, each potentially operating on very different scales. This imbalance introduces complications such as the selection of appropriate weights when scalarizing multiple objectives into a single objective function. In this paper, we propose a novel technique for scaling QUBO objectives that uses an exact computation of the variance of each individual QUBO objective. By scaling each objective to have unit variance, we align all objectives onto a common scale, thereby allowing for more balanced solutions to be found when scalarizing the objectives with equal weights, as well as potentially assisting in the search or choice of weights during scalarization. Finally, we demonstrate its advantages through empirical evaluations on various multi-objective optimization problems. Our results are noteworthy since manually selecting scalarization weights is cumbersome, and reliable, efficient solutions are scarce.

In the domain of multi-objective optimization, evolutionary algorithms are distinguished by their capability to generate a diverse population of solutions that navigate the trade-offs inherent among competing objectives. This has catalyzed the ascension of evolutionary multi-objective optimization (EMO) as a prevalent approach. Despite the effectiveness of the EMO paradigm, the analysis of resultant solution sets presents considerable challenges. This is primarily attributed to the high-dimensional nature of the data and the constraints imposed by static visualization methods, which frequently culminate in visual clutter and impede interactive exploratory analysis. To address these challenges, this paper introduces ParetoLens, a visual analytics framework specifically tailored to enhance the inspection and exploration of solution sets derived from the multi-objective evolutionary algorithms. Utilizing a modularized, algorithm-agnostic design, ParetoLens enables a detailed inspection of solution distributions in both decision and objective spaces through a suite of interactive visual representations. This approach not only mitigates the issues associated with static visualizations but also supports a more nuanced and flexible analysis process. The usability of the framework is evaluated through case studies and expert interviews, demonstrating its potential to uncover complex patterns and facilitate a deeper understanding of multi-objective optimization solution sets. A demo website of ParetoLens is available at https://dva-lab.org/paretolens/.

As the interest in multi- and many-objective optimization algorithms grows, the performance comparison of these algorithms becomes increasingly important. A large number of performance indicators for multi-objective optimization algorithms have been introduced, each of which evaluates these algorithms based on a certain aspect. Therefore, assessing the quality of multi-objective results using multiple indicators is essential to guarantee that the evaluation considers all quality perspectives. This paper proposes a novel multi-metric comparison method to rank the performance of multi-/ many-objective optimization algorithms based on a set of performance indicators. We utilize the Pareto optimality concept (i.e., non-dominated sorting algorithm) to create the rank levels of algorithms by simultaneously considering multiple performance indicators as criteria/objectives. As a result, four different techniques are proposed to rank algorithms based on their contribution at each Pareto level. This method allows researchers to utilize a set of existing/newly developed performance metrics to adequately assess/rank multi-/many-objective algorithms. The proposed methods are scalable and can accommodate in its comprehensive scheme any newly introduced metric. The method was applied to rank 10 competing algorithms in the 2018 CEC competition solving 15 many-objective test problems. The Pareto-optimal ranking was conducted based on 10 well-known multi-objective performance indicators and the results were compared to the final ranks reported by the competition, which were based on the inverted generational distance (IGD) and hypervolume indicator (HV) measures. The techniques suggested in this paper have broad applications in science and engineering, particularly in areas where multiple metrics are used for comparisons. Examples include machine learning and data mining.

Multi-objective reinforcement learning (MORL) is used to solve problems involving multiple objectives. An MORL agent must make decisions based on the diverse signals provided by distinct reward functions. Training an MORL agent yields a set of solutions (policies), each presenting distinct trade-offs among the objectives (expected returns). MORL enhances explainability by enabling fine-grained comparisons of policies in the solution set based on their trade-offs as opposed to having a single policy. However, the solution set is typically large and multi-dimensional, where each policy (e.g., a neural network) is represented by its objective values. We propose an approach for clustering the solution set generated by MORL. By considering both policy behavior and objective values, our clustering method can reveal the relationship between policy behaviors and regions in the objective space. This approach can enable decision makers (DMs) to identify overarching trends and insights in the solution set rather than examining each policy individually. We tested our method in four multi-objective environments and found it outperformed traditional k-medoids clustering. Additionally, we include a case study that demonstrates its real-world application.

This paper introduces the inverse modeling constrained multi-objective evolutionary algorithm based on decomposition (IM-C-MOEA/D) for addressing constrained real-world optimization problems. Our research builds upon the advancements made in evolutionary computing-based inverse modeling, and it strategically bridges the gaps in applying inverse models based on decomposition to problem domains with constraints. The proposed approach is experimentally evaluated on diverse real-world problems (RWMOP1-35), showing superior performance to state-of-the-art constrained multi-objective evolutionary algorithms (CMOEAs). The experimental results highlight the robustness of the algorithm and its applicability in real-world constrained optimization scenarios.