Exploration of Pareto-preserving Search Space Transformations in Multi-objective Test Functions

Many-objective optimisation, a subset of multi-objective optimisation, involves optimisation problems with more than three objectives. As the number of objectives increases, the number of solutions needed to adequately represent the entire Pareto front typically grows substantially. This makes it challenging, if not infeasible, to design a search algorithm capable of effectively exploring the entire Pareto front. This difficulty is particularly acute in the Bayesian optimisation paradigm, where sample efficiency is critical and only a limited number of solutions (often a few hundred) are evaluated. Moreover, after the optimisation process, the decision-maker eventually selects just one solution for deployment, regardless of how many high-quality, diverse solutions are available. In light of this, we argue an idea that under a very limited evaluation budget, it may be more useful to focus on finding a single solution of the highest possible quality for the decision-maker, rather than aiming to approximate the entire Pareto front as existing many-/multi-objective Bayesian optimisation methods typically do. Bearing this idea in mind, this paper proposes a single point-based multi-objective search framework (SPMO) that aims to improve the quality of solutions along a direction that leads to a good tradeoff between objectives. Within SPMO, we present a simple acquisition function, called expected single-point improvement (ESPI), working under both noiseless and noisy scenarios. We show that ESPI can be optimised effectively with gradient-based methods via the sample average approximation (SAA) approach and theoretically prove its convergence guarantees under the SAA. We also empirically demonstrate that the proposed SPMO is computationally tractable and outperforms state-of-the-arts on a wide range of benchmark and real-world problems.

Dynamic multi-objective optimization with a variable number of objectives is difficult because objective-dimensional variations may significantly change the Pareto front and degrade algorithm adaptability. This paper proposes an unbounded archive-based transfer strategy (UATS), which maintains an unbounded archive of offspring solutions within each environment stage and extracts feasible nondominated solutions as transferable elites when objective changes occur. UATS is embedded into SPEA2SDE to construct UATS-SPEA2SDE, enabling the algorithm to reuse historical evolutionary information while retaining the convergence and diversity advantages of shift-based density estimation. Experiments are conducted on four benchmark problems under three objective-changing settings, where UATS-SPEA2SDE is compared with a restart-based SPEA2SDE baseline and four representative dynamic multi-objective optimization algorithms. The results indicate that the archive-guided transfer improves recovery after environmental changes and enhances adaptability to objective-number variations.

Dynamic multi-objective optimization with a changing number of objectives has recently attracted increasing attention due to its relevance to real-world problems whose evaluation criteria may evolve over time. However, existing benchmark test suites for this problem setting suffer from a fundamental limitation: when the number of objectives changes, the objective functions themselves also change implicitly. This makes it difficult to isolate and evaluate an algorithm's capability to handle dynamics in the number of objectives alone. In this paper, we analyze this issue in detail and show that several theoretical properties claimed in prior studies rely on an assumption that is violated by commonly used test suites. To address this problem, we propose a scalable benchmark test suite in which the objective functions are fixed throughout the optimization process, while the number of active objectives changes over time. Our benchmark is constructed by defining a maximum-objective problem and dynamically selecting subsets of objectives. To avoid degeneracy issues in classical DTLZ and WFG problems, we adopt Minus-DTLZ and Minus-WFG formulations, in which all objectives are mutually conflicting. Extensive benchmark studies using representative algorithms from the literature demonstrate the usefulness and flexibility of the proposed test suite.