To address this, some works have proposed piecewise linear scalar-izations inspired by economics [Busa-Fekete et al., 2017], while for multi-armed bandits, scalarized

We study the oracle complexity of finding $\varepsilon$-Pareto stationary points in smooth multiobjective optimization with $m$ objectives. The progress metric is the Pareto stationarity gap $\mathcal{G}(x)$ (the norm of an optimal convex combination of gradients). Our contributions are fourfold. (i) For strongly convex objectives, any span first-order method (iterates lie in the span of past gradients) exhibits linear convergence no faster than $\exp(-Θ(T/\sqrtκ))$ after $T$ oracle calls, where $κ$ is the condition number, implying $Θ(\sqrtκ\log(1/\varepsilon))$ iterations; this matches classical accelerated upper bounds. (ii) For convex problems and oblivious one-step methods (a fixed scalarization with pre-scheduled step sizes), we prove a lower bound of order $1/T$ on the best gradient norm among the first $T$ iterates. (iii) Although accelerated gradient descent is outside this restricted class, it is an oblivious span method and attains the same $1/T$ upper rate on a fixed scalarization. (iv) For convex problems and general span methods with adaptive scalarizations, we establish a universal lower bound of order $1/T^{2}$ on the gradient norm of the final iterate after $T$ steps, highlighting a gap between known upper bounds and worst-case guarantees. All bounds hold on non-degenerate instances with distinct objectives and non-singleton Pareto fronts; rates are stated up to universal constants and natural problem scaling.

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.

The optimization of large-scale multibody systems is a numerically challenging task, in particular when considering multiple conflicting criteria at the same time. In this situation, we need to approximate the Pareto set of optimal compromises, which is significantly more expensive than finding a single optimum in single-objective optimization. To prevent large costs, the usage of surrogate models, constructed from a small but informative number of expensive model evaluations, is a very popular and widely studied approach. The central challenge then is to ensure a high quality (that is, near-optimality) of the solutions that were obtained using the surrogate model, which can be hard to guarantee with a single pre-computed surrogate. We present a back-and-forth approach between surrogate modeling and multi-objective optimization to improve the quality of the obtained solutions. Using the example of an expensive-to-evaluate multibody system, we compare different strategies regarding multi-objective optimization, sampling and also surrogate modeling, to identify the most promising approach in terms of computational efficiency and solution quality.

Simultaneously considering multiple objectives in machine learning has been a popular approach for several decades, with various benefits for multi-task learning, the consideration of secondary goals such as sparsity, or multicriteria hyperparameter tuning. However - as multi-objective optimization is significantly more costly than single-objective optimization - the recent focus on deep learning architectures poses considerable additional challenges due to the very large number of parameters, strong nonlinearities and stochasticity. This survey covers recent advancements in the area of multi-objective deep learning. We introduce a taxonomy of existing methods - based on the type of training algorithm as well as the decision maker's needs - before listing recent advancements, and also successful applications. All three main learning paradigms supervised learning, unsupervised learning and reinforcement learning are covered, and we also address the recently very popular area of generative modeling.

Evolutionary Multi-Objective Optimization Algorithms (EMOAs) are widely employed to tackle problems with multiple conflicting objectives. Recent research indicates that not all objectives are equally important to the decision-maker (DM). In the context of interactive EMOAs, preference information elicited from the DM during the optimization process can be leveraged to identify and discard irrelevant objectives, a crucial step when objective evaluations are computationally expensive. However, much of the existing literature fails to account for the dynamic nature of DM preferences, which can evolve throughout the decision-making process and affect the relevance of objectives. This study addresses this limitation by simulating dynamic shifts in DM preferences within a ranking-based interactive algorithm. Additionally, we propose methods to discard outdated or conflicting preferences when such shifts occur. Building on prior research, we also introduce a mechanism to safeguard relevant objectives that may become trapped in local or global optima due to the diminished correlation with the DM-provided rankings. Our experimental results demonstrate that the proposed methods effectively manage evolving preferences and significantly enhance the quality and desirability of the solutions produced by the algorithm.

Recently, there has been an increasing interest in exploring the application of multiobjective optimization (MOO) in machine learning (ML). The interest is driven by the numerous situations in real-life applications where multiple objectives need to be optimized simultaneously. A key aspect of MOO is the existence of a Pareto set, rather than a single optimal solution, which illustrates the inherent trade-offs between objectives. Despite its potential, there is a noticeable lack of satisfactory literature that could serve as an entry-level guide for ML practitioners who want to use MOO. Hence, our goal in this paper is to produce such a resource. We critically review previous studies, particularly those involving MOO in deep learning (using Physics-Informed Neural Networks (PINNs) as a guiding example), and identify misconceptions that highlight the need for a better grasp of MOO principles in ML. Using MOO of PINNs as a case study, we demonstrate the interplay between the data loss and the physics loss terms. We highlight the most common pitfalls one should avoid while using MOO techniques in ML. We begin by establishing the groundwork for MOO, focusing on well-known approaches such as the weighted sum (WS) method, alongside more complex techniques like the multiobjective gradient descent algorithm (MGDA). Additionally, we compare the results obtained from the WS and MGDA with one of the most common evolutionary algorithms, NSGA-II. We emphasize the importance of understanding the specific problem, the objective space, and the selected MOO method, while also noting that neglecting factors such as convergence can result in inaccurate outcomes and, consequently, a non-optimal solution. Our goal is to offer a clear and practical guide for ML practitioners to effectively apply MOO, particularly in the context of DL.

The competition focuses on Multiparty Multiobjective Optimization Problems (MPMOPs), where multiple decision makers have conflicting objectives, as seen in applications like UAV path planning. Despite their importance, MPMOPs remain understudied in comparison to conventional multiobjective optimization. The competition aims to address this gap by encouraging researchers to explore tailored modeling approaches. The test suite comprises two parts: problems with common Pareto optimal solutions and Biparty Multiobjective UAV Path Planning (BPMO-UAVPP) problems with unknown solutions. Optimization algorithms for the first part are evaluated using Multiparty Inverted Generational Distance (MPIGD), and the second part is evaluated using Multiparty Hypervolume (MPHV) metrics. The average algorithm ranking across all problems serves as a performance benchmark.

In industry, Bayesian optimization (BO) is widely applied in the human-AI collaborative parameter tuning of cyber-physical systems. However, BO's solutions may deviate from human experts' actual goal due to approximation errors and simplified objectives, requiring subsequent tuning. The black-box nature of BO limits the collaborative tuning process because the expert does not trust the BO recommendations. Current explainable AI (XAI) methods are not tailored for optimization and thus fall short of addressing this gap. To bridge this gap, we propose TNTRules (TUNE-NOTUNE Rules), a post-hoc, rule-based explainability method that produces high quality explanations through multiobjective optimization. Our evaluation of benchmark optimization problems and real-world hyperparameter optimization tasks demonstrates TNTRules' superiority over state-of-the-art XAI methods in generating high quality explanations. This work contributes to the intersection of BO and XAI, providing interpretable optimization techniques for real-world applications.