OPAL: Operator-Programmed Algorithms for Landscape-Aware Black-Box Optimization

1 OPAL: Operator-Programmed Algorithms for Landscape-Aware Black-Box Optimization Junbo Jacob Lian, Mingyang Yu, Kaichen Ouyang, Shengwei Fu, Rui Zhong, Yujun Zhang, Jun Zhang, Fellow, IEEE, and Huiling Chen, Member, IEEE Abstract—Black-box optimization often relies on evolutionary and swarm algorithms whose performance is highly problem dependent. We view an optimizer as a short program over a small vocabulary of search operators and learn this operator program separately for each problem instance. We instantiate this idea in Operator-Programmed Algorithms (OPAL), a landscape- aware framework for continuous black-box optimization that uses a small design budget with a standard differential evolution baseline to probe the landscape, builds a k-nearest neighbor graph over sampled points, and encodes this trajectory with a graph neural network. A meta-learner then maps the resulting representation to a phase-wise schedule of exploration, restart, and local search operators. On the CEC 2017 test suite, a single meta-trained OPAL policy is statistically competitive with state- of-the-art adaptive differential evolution variants and achieves significant improvements over simpler baselines under nonpara- metric tests. Ablation studies on CEC 2017 justify the choices for the design phase, the trajectory graph, and the operator- program representation, while the meta-components add only modest wall-clock overhead. Overall, the results indicate that operator-programmed, landscape-aware per-instance design is a practical way forward beyond ad hoc metaphor-based algorithms in black-box optimization. Index Terms—Black-box optimization, evolutionary computa- tion, hyper-heuristics, meta-learning, graph neural networks. I. INTRODUCTION C ONTINUOUS black-box optimization (BBO) lies at the core of many engineering, control, and machine learning applications. In these settings, the objective func- tion is expensive, derivative-free, noisy, or multimodal, and This research is financially supported by the National Natural Science Foun- dation of China (Grant No. 62076185, 62301367). (Corresponding authors: Jun Zhang & Huiling Chen.) Junbo Jacob Lian is with the McCormick School of Engi- neering, Northwestern University, Evanston, IL, USA (e-mail: Jacoblian@u.northwestern.edu). Mingyang Yu is with the College of Artificial Intelligence, Nankai Univer- sity, Tianjin, China (e-mail: 1120240312@mail.nankai.edu.cn). Kaichen Ouyang is with the School of Mathematics, University of Science and Technology of China, Hefei, China (e-mail: oykc@mail.ustc.edu.cn). Shengwei Fu is with Guizhou University, Guiyang, China (e-mail: gs.swfu22@gzu.edu.cn). Yujun Zhang is with the School of New Energy, Jingchu University of Technology, Jingmen, China (e-mail: zhangyj069@gmail.com). Rui Zhong is with the Information Initiative Center, Hokkaido University, Sapporo, Japan (e-mail: zhongrui@iic.hokudai.ac.jp). Jun Zhang is with the College of Artificial Intelligence, Nankai University, Tianjin, China (e-mail: junzhang@nankai.edu.cn). Huiling Chen is with the School of Computer Science and Artificial Intelligence, Wenzhou University, Wenzhou, China (e-mail: chenhuiling.jlu@gmail.com). Source code, experiment scripts, and results are publicly available at https: //github.com/junbolian/OPAL. practitioners typically rely on population-based metaheuris- tics such as differential evolution (DE) and particle swarm optimization (PSO) [1]–[3]. Over the last two decades, the field has produced a large ecosystem of increasingly so- phisticated variants—for example L-SHADE [4] and jSO [5] in the DE family [6]—that achieve strong performance on standard benchmarks but remain highly problem-dependent. The No Free Lunch theorems for optimization make this dependence formal: averaged over all possible problems, no single algorithm or configuration can dominate all others [7]. In practice, even within a single problem instance, algorithmic needs change over time as the search progresses from global exploration to local exploitation [8]. A natural response has been to use machine learning to support or replace human design. One line of work builds per-instance algorithm selectors and configurators: a feature extractor characterizes the problem, and a model predicts which algorithm or parameter setting will perform best [9], [10]. Exploratory landscape analysis (ELA) plays a central role in this program: it maps sampled points into numerical descriptors of modality, ruggedness, conditioning, and other properties, which are then fed into supervised models for solver selection or performance prediction [10], [11]. Recent surveys document a rapid growth in such ML-assisted meta- heuristics, covering both single-objective and multiobjective settings and highlighting hybrid designs where learned models steer classical metaheuristics [12]. A second line of work treats the optimizer itself as a dynamical system and learns to adapt its parameters on- line. Automated and dynami

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