Efficiently Tackling Million-Dimensional Multiobjective Problems: A Direction Sampling and Fine-Tuning Approach

TL;DR

The paper introduces VMOF, a framework for very large-scale multiobjective optimization with over $10^5$ decision variables. It combines Thompson sampling to identify promising evolutionary directions in a high-dimensional space with a per-solution direction fine-tuning process to closely approximate the Pareto frontier under limited function evaluations. The approach is analyzed theoretically and validated empirically against state-of-the-art MOEAs on VLSMOP and LSMOP benchmarks as well as real-world TREE problems, showing superior convergence and diversity in most settings. The work demonstrates that sampling global directions and then refining them locally can effectively mitigate the curse of dimensionality and enable scalable optimization in practical, high-dimensional applications.

Abstract

We define very large-scale multiobjective optimization problems as optimizing multiple objectives (VLSMOPs) with more than 100,000 decision variables. These problems hold substantial significance, given the ubiquity of real-world scenarios necessitating the optimization of hundreds of thousands, if not millions, of variables. However, the larger dimension in VLSMOPs intensifies the curse of dimensionality and poses significant challenges for existing large-scale evolutionary multiobjective algorithms, rendering them more difficult to solve within the constraints of practical computing resources. To overcome this issue, we propose a novel approach called the very large-scale multiobjective optimization framework (VMOF). The method efficiently samples general yet suitable evolutionary directions in the very large-scale space and subsequently fine-tunes these directions to locate the Pareto-optimal solutions. To sample the most suitable evolutionary directions for different solutions, Thompson sampling is adopted for its effectiveness in recommending from a very large number of items within limited historical evaluations. Furthermore, a technique is designed for fine-tuning directions specific to tracking Pareto-optimal solutions. To understand the designed framework, we present our analysis of the framework and then evaluate VMOF using widely recognized benchmarks and real-world problems spanning dimensions from 100 to 1,000,000. Experimental results demonstrate that our method exhibits superior performance not only on LSMOPs but also on VLSMOPs when compared to existing algorithms.

Figures (7)

• Figure 1: Illustration of the proposed very large-scale multiobjective optimization framework (VMOF). 1. Initialize a set of directions and solutions and randomly partition them into several groups. 2. Sample and recommend a direction for each group. 3. Fine-tune directions for all solutions based on recommended directions. 4. Evolve each solution according to fine-tuned directions.
• Figure 2: This illustration depicts the process of evolution direction sampling based on Thompson sampling. Initially, each individual possesses a unique direction ($\boldsymbol d_i$) within the decision space, guiding the individual through a single evolution. Subsequently, each individual ($\boldsymbol p_i$) undergoes evolution and evaluation within the objective space. Based on the evaluation results, the algorithm updates the corresponding $\alpha$ and $\beta$ parameters for each direction. Notably, $\alpha$ and $\beta$ represent the number of times an individual has evolved and degenerated along direction $\boldsymbol d_i$, respectively. The algorithm then calculates the $P^{Beta}(\theta)$ distribution for each direction (represented by blue lines in $P^{Beta}_i$), and samples the mean reward $\bar{\theta}_i$ from the $P^{Beta}(\theta)$ distribution (indicated by the red lines in $P^{Beta}_i$). The direction corresponding to the distribution with the highest mean reward $\bar{\theta}_i$ is selected. In this example, the sampled direction is $\boldsymbol d_1$, as determined by $P^{Beta}_1$.
• Figure 3: Illustration of evolution directions fine-tuning. 1. For paired group of solutions $\mathcal{S}_i$ and group of directions $\mathcal{D}_d$. 2. Initialize a set of direction populations based on the recommended direction, then evolve each solution with direction population and evaluate; 4. Obtain a set of directions for each solution.
• Figure 4: Nondominated solutions obtained by compared methods on TREE with 100,000 decision variables. Please refer to the supplementary materials for other algorithms.
• Figure 5: Convergence profiles of all compared algorithms on bi-objective LSMOP1 with 1,000 and 100,000 decision variables, respectively.