This paper presents a principled framework for designing energy-aware metaheuristics that operate under fixed energy budgets. We introduce a unified operator-level model that quantifies both numerical gain and energy usage, and define a robust Expected Improvement per Joule (EI/J) score that guides adaptive selection among operator variants during the search. The resulting energy-aware solvers dynamically choose between operators to self-control exploration and exploitation, aiming to maximize fitness gain under limited energy. We instantiate this framework with three representative metaheuristics - steady-state GA, PSO, and ILS - each equipped with both lightweight and heavy operator variants. Experiments on three heterogeneous combinatorial problems (Knapsack, NK-landscapes, and Error-Correcting Codes) show that the energy-aware variants consistently reach comparable fitness while requiring substantially less energy than their non-energy-aware baselines. EI/J values stabilize early and yield clear operator-selection patterns, with each solver reliably self-identifying the most improvement-per-Joule - efficient operator across problems.

翻译：本文提出了一种在固定能量预算下设计能量感知元启发式算法的原则性框架。我们引入了一个统一的算子级模型，该模型同时量化数值增益与能量消耗，并定义了一个稳健的每焦耳期望改进（EI/J）指标，用于指导搜索过程中不同算子变体的自适应选择。由此产生的能量感知求解器能够在算子间动态选择，以实现对探索与开发的自主调控，目标是在有限能量下最大化适应度增益。我们通过三种代表性元启发式算法——稳态遗传算法（GA）、粒子群优化（PSO）和迭代局部搜索（ILS）——实例化了该框架，每种算法均配备了轻量级和重量级两种算子变体。在三个异构组合问题（背包问题、NK景观和纠错码）上的实验表明，能量感知变体在达到可比适应度的同时，所需能量显著低于非能量感知基线。EI/J值早期即趋于稳定，并产生清晰的算子选择模式，每种求解器都能可靠地自我识别出跨问题中每焦耳改进效率最高的算子。