Optimization problems are prevalent across various scenarios. Formulating and then solving optimization problems described by natural language often requires highly specialized human expertise, which could block the widespread application of optimization-based decision making. To automate problem formulation and solving, leveraging large language models (LLMs) has emerged as a potential way. However, this kind of approach suffers from the issue of optimization generalization. Namely, the accuracy of most current LLM-based methods and the generality of optimization problem types that they can model are still limited. In this paper, we propose a unified learning-based framework called LLMOPT to boost optimization generalization. Starting from the natural language descriptions of optimization problems and a pre-trained LLM, LLMOPT constructs the introduced five-element formulation as a universal model for learning to define diverse optimization problem types. Then, LLMOPT employs the multi-instruction tuning to enhance both problem formalization and solver code generation accuracy and generality. After that, to prevent hallucinations in LLMs, such as sacrificing solving accuracy to avoid execution errors, the model alignment and self-correction mechanism are adopted in LLMOPT. We evaluate the optimization generalization ability of LLMOPT and compared methods across six real-world datasets covering roughly 20 fields such as health, environment, energy and manufacturing, etc. Extensive experiment results show that LLMOPT is able to model various optimization problem types such as linear/nonlinear programming, mixed integer programming, and combinatorial optimization, and achieves a notable 11.08% average solving accuracy improvement compared with the state-of-the-art methods. The code is available at https://github.com/caigaojiang/LLMOPT.

Black-box optimization aims to find the optima through building a model close to the black-box objective function based on function value evaluation. However, in many real-world tasks, such as the design of molecular formulas and mechanical structures, it is perilous, costly, or even infeasible to evaluate the objective function value of an actively sampled solution. In this situation, optimization can only be conducted via utilizing offline historical data, which yields offline black-box optimization. Different from the traditional goal that is to pursue the optimal solution, this paper emphasizes that the goal of offline optimization is to stably surpass the offline dataset during optimization procedure. Although benchmarks called Design-Bench already exist in this emerging field, it can hardly evaluate the stability of offline optimization and mainly provides real-world offline tasks and the corresponding offline datasets. To this end, this paper proposes benchmarks named SOO-Bench (i.e., Stable Offline Optimization Benchmarks) for offline black-box optimization algorithms, so as to systematically evaluate the stability of surpassing the offline dataset under different data distributions. Along with SOO-Bench, we also propose a stability indicator to measure the degree of stability. Specifically, SOO-Bench includes various real-world offline optimization tasks and offline datasets under different data distributions, involving the fields of satellites, materials science, structural mechanics, and automobile manufacturing. Empirically, baseline and state-of-the-art algorithms are tested and analyzed on SOO-Bench. Hopefully, SOO-Bench is expected to serve as a catalyst for the rapid developments of more novel and stable offline optimization methods. The code is available at \url{https://github.com/zhuyiyi-123/SOO-Bench}.

In black-box optimization, when directly evaluating the function values of solutions is very costly or infeasible, access to the objective function is often limited to comparing pairs of solutions, which yields dueling black-box optimization. Dueling optimization is solely based on pairwise preferences, and thus notably reduces cost compared with function value based methods. However, the optimization performance of dueling optimization is often limited due to that most existing dueling optimization methods do not make full use of the pairwise preferences collected. To better utilize these preferences, this paper proposes relation-augmented dueling Bayesian optimization (RADBO) via preference propagation. By considering solution similarity, RADBO aims to uncover the potential dueling relations between solutions within different preferences through the proposed preference propagation technique. Specifically, RADBO first clusters solutions using a Gaussian mixture model. After obtaining the solution set with the highest intra-cluster similarity, RADBO utilizes a directed hypergraph to model the potential dueling relations between solutions, thereby realizing relation augmentation. Extensive experiments are conducted on both synthetic functions and real-world tasks such as motion control, car cab design and spacecraft trajectory optimization. The experimental results disclose the satisfactory accuracy of augmented preferences in RADBO, and show the superiority of RADBO compared with existing dueling optimization methods. Notably, it is verified that, under the same evaluation cost budget, RADBO can be competitive with or even surpass the function value based Bayesian optimization methods with respect to optimization performance.

In many scenarios of black-box optimization, evaluating the objective function values of solutions is expensive, while comparing a pair of solutions is relatively cheap, which yields the dueling black-box optimization. The side effect of dueling optimization is that it doubles the dimension of solution space and exacerbates the dimensionality scalability issue of black-box optimization, e.g., Bayesian optimization. To address this issue, the existing dueling optimization methods fix one solution when dueling throughout the optimization process, but it may reduce their efficacy. Fortunately, it has been observed that, in recommendation systems, the dueling results are mainly determined by the latent human preferences. In this paper, we abstract this phenomenon as the preferential intrinsic dimension and inject it into the dueling Bayesian optimization, resulting in the preferential embedding dueling Bayesian optimization (PE-DBO). PE-DBO decouples optimization and pairwise comparison via the preferential embedding matrix. Optimization is performed in the preferential intrinsic subspace with much lower dimensionality, while pairwise comparison is completed in the original dueling solution space. Theoretically, we disclose that the preference function can be approximately preserved in the lower-dimensional preferential intrinsic subspace. Experiment results verify that, on molecule discovery and web page recommendation dueling optimization tasks, the preferential intrinsic dimension exists and PE-DBO is superior in scalability compared with that of the state-of-the-art (SOTA) methods.