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Minimal MMAO: A Resource-Closed-Loop Framework for Adaptive Metaheuristic Search
Jinliang Xu, Liping Ma
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 98%
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Summary
The paper introduces the Metabolic Multi-Agent Optimizer (MMAO), an adaptive metaheuristic framework based on an endogenous resource-closed-loop system. Unlike traditional adaptive metaheuristics that use external schedules, MMAO regulates search intensity, exploration-exploitation balance, and lifecycle turnover through a shared metabolic controller involving bounded private energy, a communal budget, and normalized rewards. The framework was instantiated in both continuous (e.g., Sphere, Rastrigin) and discrete (e.g., Euclidean TSP, TSPLIB) domains. Results indicate that while the metabolic loop is domain-agnostic and stable, continuous optimization requires richer realization for high precision, whereas discrete optimization is more tolerant of compact, lean implementations.
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Relation Signals (5)
Jinliang Xu → authored → Metabolic Multi-Agent Optimizer
confidence 100% · Minimal MMAO: A Resource-Closed-Loop Framework for Adaptive Metaheuristic Search Jinliang Xu ∗ and Liping Ma
Metabolic Multi-Agent Optimizer → evaluatedon → Sphere
confidence 100% · The method is then instantiated in both continuous and discrete domains and evaluated on a matched small-scale suite including Sphere, Rastrigin, a synthetic Euclidean TSP, and two TSPLIB instances.
Metabolic Multi-Agent Optimizer → evaluatedon → Rastrigin
confidence 100% · The method is then instantiated in both continuous and discrete domains and evaluated on a matched small-scale suite including Sphere, Rastrigin, a synthetic Euclidean TSP, and two TSPLIB instances.
Metabolic Multi-Agent Optimizer → usesprinciple → Bounded Private Energy
confidence 100% · We formulate MMAO through bounded private energy, a communal budget, normalized reward, continuous role adaptation, and resource-financed branching and pruning.
Metabolic Multi-Agent Optimizer → usesprinciple → Communal Budget
confidence 100% · We formulate MMAO through bounded private energy, a communal budget, normalized reward, continuous role adaptation, and resource-financed branching and pruning.
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Abstract
Abstract:This paper presents the Metabolic Multi-Agent Optimizer (MMAO) as an adaptive metaheuristic built around endogenous resource circulation. The central premise is that search intensity, exploration--exploitation balance, and lifecycle turnover should be induced by a shared metabolic controller rather than by separately attached schedules. We formulate MMAO through bounded private energy, a communal budget, normalized reward, continuous role adaptation, and resource-financed branching and pruning. The method is then instantiated in both continuous and discrete domains and evaluated on a matched small-scale suite including Sphere, Rastrigin, a synthetic Euclidean TSP, and two TSPLIB instances. The results show a consistent pattern: the same metabolic loop remains workable across domains, the discrete realization remains relatively stable under a compact design, and continuous refinement quality is the main cost of keeping the method lean. Taken together, these findings position MMAO as a coherent framework for adaptive heuristic design rather than a loose collection of operators.
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- Source: https://arxiv.org/abs/2606.30450v1
- Canonical: https://arxiv.org/abs/2606.30450v1
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1 Minimal MMAO: A Resource-Closed-Loop Framework for Adaptive Metaheuristic Search Jinliang Xu ∗ and Liping Ma Abstract—This paper presents the Metabolic Multi-Agent Op- timizer (MMAO) as an adaptive metaheuristic built around endogenous resource circulation. The central premise is that search intensity, exploration–exploitation balance, and lifecycle turnover should be induced by a shared metabolic controller rather than by separately attached schedules. We formulate MMAO through bounded private energy, a communal budget, normalized reward, continuous role adaptation, and resource- financed branching and pruning. The method is then instantiated in both continuous and discrete domains and evaluated on a matched small-scale suite including Sphere, Rastrigin, a synthetic Euclidean TSP, and two TSPLIB instances. The results show a consistent pattern: the same metabolic loop remains workable across domains, the discrete realization remains relatively stable under a compact design, and continuous refinement quality is the main cost of keeping the method lean. Taken together, these findings position MMAO as a coherent framework for adaptive heuristic design rather than a loose collection of operators. Index Terms—Adaptive metaheuristics, endogenous resource allocation, exploration–exploitation control, multi-agent opti- mization, parameter-light optimization. I. INTRODUCTION Modern heuristic search increasingly relies on adaptive components such as dynamic populations, online operator selection, self-adjusting step scales, and budget reallocation across subpopulations or tasks. These developments are useful, but they also raise a recurring methodological problem. As algorithms become more adaptive, the organizing principle behind that adaptation often becomes less explicit. It then becomes difficult to tell whether a method is a coherent framework or only a successful bundle of mechanisms [1]– [4]. This paper focuses on a different organizing principle, namely metabolic control. Instead of prescribing separate schedules for search radius, exploitation intensity, population size, and reinvestment, MMAO lets these behaviors emerge from a resource-closed loop shared by all agents. Each agent maintains bounded private energy, successful search contributes to a communal pool, normalized reward converts problem-dependent progress into comparable signals, and life- cycle events are paid for from the same budget that sustains search. In this view, the algorithm does not merely apply operators; it maintains a self-regulating economy of search effort. The central question of this paper is whether such a formu- lation can remain informative and operational when written in Jinliang Xu is with the current study team; e-mail: jlxufly@gmail.com. Liping Ma is with the Department of Disease Control and Prevention, The Seventh Medical Center of Chinese PLA General Hospital, Beijing, China; e-mail: lipingmaqzx@163.com. a compact, explicit form. The objective is not to maximize benchmark coverage, but to expose the framework identity clearly enough that later theoretical, derivative, or large-scale experimental work can build on it without ambiguity. We therefore emphasize three goals: 1) formulate the essential MMAO state variables and re- source flows in a domain-agnostic way; 2) give continuous and discrete instantiations driven by the same metabolic loop; 3) evaluate the empirical behavior of the framework under matched tasks and budgets, so that its strengths and limitations are visible rather than implied. The main contributions of this paper are as follows. 1) We formalize MMAO as a compact adaptive meta- heuristic framework based on five coupled principles: bounded private energy, a communal pool, normalized reward, continuous role dynamics, and resource-financed lifecycle turnover. 2) We provide domain-specific realizations for continuous and permutation search while preserving a single shared control logic. 3) We present a mechanism mapping that distinguishes framework-essential components from optional instan- tiation detail. 4) We report a matched small-scale experimental study across four continuous tasks and three TSP tasks, in- cluding TSPLIB eil51 and berlin52, together with budget-fairness explanations and representative conver- gence traces. I. RELATED WORK MMAO is most closely related to research on endogenous parameter control, resource allocation, and adaptive popula- tion management. In evolutionary computation, the distinction between exogenous parameter schedules and endogenous self- adaptation has long been emphasized [1], [2], [5], [6]. More recent work has broadened this perspective toward dynamic population sizes, budget redistribution, and online learning of search behavior [7]–[12]. From the continuous-optimization side, MMAO also con- nects to derivative-free search and self-adaptive refinement. Symmetric probing and progress-normalized local sensing resemble a lightweight zero-order view of search, while con- tinuous role modulation provides an endogenous alternative to explicit mode switching [13]–[15]. From the discrete side, the route-construction and route-improvement instantiations place MMAO near structure-aware TSP heuristics, though the arXiv:2606.30450v1 [cs.NE] 29 Jun 2026 2 present goal is not to compete with highly specialized methods such as Lin–Kernighan variants or strong neural-improvement pipelines [16]–[20]. What distinguishes MMAO from these lines of work is not any single operator. Rather, it is the insistence that adaptation should be derived from a closed metabolic budget. Search, reinforcement, and turnover are not treated as loosely coupled modules; they are consequences of one common resource economy. That framework viewpoint is the main object of study here. I. MMAO FRAMEWORK FORMULATION A. State and Resource Loop Let P t = A 1 ,...,A N t denote the active population at iteration t. Each agent carries the state A i (t) = x i (t),E i (t),φ i (t),m i (t) ,(1) where x i (t) is the current candidate solution, E i (t) ∈ [0,E max ] is bounded private energy, φ i (t) ∈ [0, 1] is a continuous role variable, and m i (t) is local memory such as a personal best or structural elite route. At the population level, the framework maintains a com- munal budget B t and a progress scale s t . Raw improvement ∆ i (t) is mapped into normalized gain g i (t) = max(0, ∆ i (t)) s t + ε ,(2) so that reward reacts to relative progress rather than to the absolute magnitude of the objective function. The metabolic loop is then expressed abstractly as E i (t + 1) = Π [0,E max ] (E i (t) + (1− α)R(g i (t))− C i (t)), (3) B t+1 = max (0,B t + αR(g i (t))− U t ),(4) where R(·) is a bounded reward transform, α is the communal share, C i (t) is private maintenance, and U t is the resource spent on branching, pruning, or respawn. In this formulation, adaptation is not attached externally. It is induced by changes in gain, energy, role, and communal budget. B. Framework Principles The present study treats the following five principles as framework-defining. • Bounded private energy: each agent has a finite local budget and cannot search indefinitely at full intensity. • Communal pool: part of successful progress is socialized and can later be reinvested elsewhere in the population. • Normalized reward: adaptation depends on scale-aware progress signals rather than raw objective differences. • Continuous role dynamics: agents move on an exploration–exploitation continuum rather than belonging to hard-coded species. • Lifecycle turnover: branching, pruning, and respawn are paid for by the same resource loop that governs search itself. These principles are illustrated in Fig. 1, which shows the closed interaction between state, sensing, action, evaluation, metabolism, and turnover. Agent state x i , E i , φ i , m i , τ i Domain-aware sensing symmetric probingorstructural sensing Action and local refinement role-interpolated moveorroute modification Objective evaluation and normalized gain ∆f i → g i (t) using progress scale s t Metabolic update private energy, communal pool B t , maintenance, tax, reward Redistribution and turnover branch, prune, respawn, elite reinvestment Global metabolic state B t : communal pool s t : progress scale ρ t : success rate spread / edge memory role and budget con- trol Search loop Metabolic controller shared controller across domains Fig. 1. Resource-closed-loop architecture of MMAO. Search behavior, reward, and lifecycle turnover are all regulated by the same metabolic controller. IV. CONTINUOUS AND DISCRETE INSTANTIATIONS A. Continuous Instantiation In the continuous setting, each agent performs symmetric local sensing around its current state, forms a role-conditioned candidate move by combining probe information with global and personal guidance, and then updates its energy and role according to normalized gain. Agents with sustained gains can finance local offspring around promising anchors, while de- pleted agents are pruned and the communal pool may support respawn. The implementation therefore combines zero-order sensing, lightweight memory, and lifecycle control without introducing a separate external scheduler. B. Discrete Instantiation In the permutation setting, the same loop is realized through sampled 2-opt improvement, route kick/mutation, and lightweight edge-memory reconstruction. Role values regu- late the balance between local route exploitation and more disruptive route modification, while the communal budget controls whether high-performing routes seed new candidates. This design aims to preserve structural awareness without collapsing into a task-specific handcrafted TSP solver. C. Mechanism Mapping To clarify what belongs to the framework and what belongs only to a particular realization, Table I organizes MMAO by control responsibility rather than by an unstructured list of operators. The main point is that the framework is defined by the metabolic loop and its control obligations, not by any one motion equation or route operator. V. ALGORITHMIC PROCEDURE Algorithm 1 summarizes the common MMAO loop. Domain-specific proposal and evaluation routines differ, but the metabolic controller is shared. Figure 2 provides an intuitive view of why the framework differs from a static equal-effort population: resources are 3 TABLE I LAYERED MECHANISM VIEW OF MMAO. FRAMEWORK RESPONSIBILITIES ARE SEPARATED FROM DOMAIN-SPECIFIC REALIZATIONS. LayerControl itemFramework responsibilityRealization in this paper Metabolic statePrivate energyBounded local effort and survival stateKept as agent-level energy reserve Metabolic stateCommunal poolShared reinvestment budget across agentsKept as population-level resource account Metabolic stateNormalized rewardMakes progress comparable across scalesKept through rolling gain normalization Adaptive behaviorContinuous role stateInterpolates exploration and exploitationKept as a continuous role variable Adaptive behaviorLifecycle turnoverFinances branch, prune, and respawn from the same loop Kept as budget-driven turnover Continuous realization Symmetric probingProvides local zero-order sensingUsed in compact probe-based form Continuous realization Personal/global couplingBlends memory and population guidanceUsed in compact role-conditioned moves Continuous realization Elite precision refinementExtends late-stage search sharpnessNot emphasized in the current compact setup Discrete realizationSampled 2-opt improvementProvides route-level local refinementUsed directly Discrete realizationRoute kick / mutationSupports escape from route stagnationUsed directly Discrete realizationEdge memory biasReuses shared structural informationUsed in lightweight form Discrete realizationExtra reinvestment heuristicsStrengthens route reconstruction detailNot emphasized in the current compact setup Algorithm 1: Minimal MMAO Framework Input: Objective or cost function f , search domain Ω, initial population P 0 , initial communal budget B 0 , iteration limit T Output: Best solution found x ⋆ Initialize agent states (x i ,E i ,φ i ,m i ) and communal budget B 0 ; Estimate initial progress scale s 0 ; for t = 0 to T − 1 do foreach agent A i ∈P t do Perform domain-aware sensing around x i (t); Generate one or more role-conditioned candidate states; Evaluate candidates and compute raw improvement ∆ i (t); Convert ∆ i (t) into normalized gain g i (t); Update private energy E i (t + 1) and communal budget B t+1 ; Update role state φ i (t + 1) from gain and energy pressure; Refresh local memory m i (t + 1) and global best if improved; end Prune agents with exhausted energy if population is above the minimum size; Reinvest communal resources into branching or respawn when budget and parent quality permit; Update progress scale s t from recent positive gains; end Return the best solution encountered; allowed to concentrate around productive regions, but only through endogenous gain and budget feedback. VI. EXPERIMENTAL PROTOCOL A. Tasks and Evaluation The purpose of the experiments is to test whether the frame- work behaves coherently and meaningfully in both continuous Static Population MMAO Redistribution equal effortadaptive reinvestment Fig. 2.Conceptual contrast between a static equal-effort population and MMAO-style adaptive redistribution. and discrete domains. We therefore use a matched small-scale suite rather than a large benchmark campaign. The continuous side contains Sphere-2D, Rastrigin-2D, Sphere-5D, and Rastrigin-5D. These tasks provide a pro- gression from smooth unimodal landscapes to more multi- modal and dimensionally demanding settings. The discrete side contains a synthetic 10-city Euclidean TSP together with TSPLIB eil51 and berlin52. For TSPLIB instances, we report gap to the known optimum. For the synthetic 10- city instance, we compute the exact optimum by Held–Karp dynamic programming. B. Budget Fairness The goal of the study is not to claim superiority over unrelated optimizers, but to assess the behavior of the MMAO framework under a consistent resource envelope. Fairness is therefore enforced in three ways. • Both continuous and discrete instantiations use the same five seeds 3, 7, 11, 19, 23 on every task. • For each task, the compared variants use the same it- eration budget and the same minimum/initial/maximum population limits. • The only intended difference is the level of mechanism detail inside the same metabolic controller, not the exter- nal search budget. This design should be interpreted as within-framework fair- ness. It supports diagnosis of the framework behavior itself, not broad benchmarking against unrelated algorithm families. 4 TABLE I CONTINUOUS-TASK SUMMARY. LOWER IS BETTER. TaskIterMeanStdBest Sphere-2D (MMAO)1200.0000010.0000010.000000 Sphere-2D (compact)1200.0067780.0052010.001184 Rastrigin-2D (MMAO)1400.1993740.3980560.000180 Rastrigin-2D (compact)1400.4004220.3134150.110167 Sphere-5D (MMAO)1800.0000030.0000000.000002 Sphere-5D (compact)1800.0245990.0130850.010674 Rastrigin-5D (MMAO)2201.7914630.7445840.995365 Rastrigin-5D (compact)2208.0783350.9714236.775656 TABLE I TSP SUMMARY. LOWER GAP IS BETTER. TaskIterMean Gap (%) MMAOMean Gap (%) compactBest compact Synthetic-101600.000.0027.9274 eil512603.054.84433 berlin522804.586.087547 VII. EXPERIMENTAL RESULTS A. Summary Tables Tables I and I summarize the results. On the contin- uous tasks, MMAO remains functional in all settings, but the more compact realization gives up a visible amount of final precision. The effect is mild on easy Sphere cases and becomes much stronger on Rastrigin-5D, where the mean best fitness rises from 1.7915 to 8.0783. This indicates that the common metabolic loop is sufficient for stable behavior, but rich late-stage refinement remains important when the landscape becomes more multimodal. On the discrete tasks, the picture is more stable. Both vari- ants solve the synthetic 10-city problem optimally. On eil51 and berlin52, the compact realization incurs moderate gap increases, yet the degradation is far smaller than on the difficult continuous case. This suggests that the resource-closed-loop idea transfers well to structure-aware permutation search even when the realization remains relatively lean. B. Representative Convergence Behavior Representative convergence traces on Rastrigin-5D and eil51 are shown in Fig. 3. The continuous trace highlights that the richer instantiation keeps improving later into the search, whereas the compact variant stabilizes earlier. By contrast, the discrete trace shows a much smaller separation between the two curves, which is consistent with the gap statistics in Table I. C. Extended Analysis The continuous and discrete sides expose different conse- quences of using the same closed-loop control principle. On the continuous side, the main challenge is not merely to find improvement, but to preserve sufficiently fine-grained adapta- tion once the population has already concentrated. This is why the difference is small in terms of qualitative convergence, yet large in terms of final precision. On Sphere, both variants enter the attraction basin reliably, but the compact realization 050100150200 100 150 200 250 300 350 050100150200250 100 150 200 250 300 350 Fig. 3. Representative mean convergence traces on Rastrigin-5D (top) and eil51 (bottom). already loses several orders of magnitude of terminal accuracy. On Rastrigin, especially in 5D, the gap widens further because multimodality makes later reinvestment quality and proposal sharpness much more important. The discrete side is more forgiving. Once route-level local improvement, occasional disruptive kicks, and shared edge- memory bias are all present, the same metabolic controller can redistribute effort effectively without needing many additional task-specific mechanisms. This is most obvious in the synthetic 10-city case, where the compact realization preserves exact- quality behavior. It remains true, though to a lesser extent, on eil51 and berlin52. The compact version degrades, but the relative increase in mean objective remains modest compared with the continuous degradation on Rastrigin-5D. This asymmetry matters for how the framework should be interpreted. The present evidence suggests that MMAO’s primary value is not that one compact realization dominates all settings, but that the same resource economy provides a stable adaptive backbone across domains. That backbone can then be paired with stronger domain-aware realizations when higher final precision is required. D. Interpretation Three observations emerge from the current experiments. First, the same resource loop remains workable across both domains without changing its organizing principle. Second, the framework is expressive enough that a richer realization can purchase better continuous refinement without changing the high-level logic. Third, route-based permutation search appears more tolerant to compact realization than multimodal continuous refinement in the current suite. 5 These observations should be interpreted cautiously. The experiments are intentionally small and diagnostic, not exhaus- tive. Their value lies in exposing a coherent behavior pattern rather than in claiming universal superiority. VIII. DISCUSSION The present results support MMAO as a meaningful frame- work proposal, but they also indicate where the main research pressure points lie. First, the framework seems strongest when one cares about how adaptation is generated. MMAO offers a single explana- tory language for search intensity, role change, reinvestment, and turnover. This is a methodological advantage over designs where each adaptive component is added independently. Second, the continuous results show that compactness and final precision can be in tension. This is not a weakness of the framework idea itself, but it does imply that future high-performance instantiations will need better late-stage refinement while still keeping the resource loop central. Third, the discrete results suggest that structure-aware local improvement and shared memory can coexist naturally with the metabolic controller. That makes combinatorial optimiza- tion a promising direction for future derivative MMAO studies. Finally, the framework view clarifies a useful separation of concerns: the metabolic loop defines the family identity, whereas operators and proposal models define the domain- specific realization. This separation should make later deriva- tive studies easier to organize, compare, and interpret. A. Limitations and Future Work Several limitations should be stated explicitly. First, the current empirical study is intentionally small and diagnostic; it is sufficient for framework-level interpretation, but not for strong competitiveness claims. Second, the continuous side currently shows that compact realization can sacrifice a nontrivial amount of late-stage precision, especially on multimodal landscapes. Third, the discrete evidence is limited to routing problems and should not yet be generalized to all combinatorial domains. These limitations point directly to the next research steps. The first is to design stronger continuous proposal and re- finement mechanisms that remain metabolically derived rather than externally attached. The second is to extend the empir- ical study to larger benchmark suites and richer statistical treatment. The third is to test whether the same framework logic transfers to other discrete structures such as assignment and scheduling. A longer-term direction is formal analysis of the communal-budget dynamics themselves, especially how normalized reward and role evolution interact to produce stable redistribution. IX. CONCLUSION This paper presented MMAO as a resource-closed-loop framework for adaptive heuristic search. The framework is built around bounded private energy, a communal resource pool, normalized reward, continuous role adaptation, and resource-financed lifecycle turnover. A single control logic was instantiated in continuous and discrete search spaces and tested on a matched small-scale suite. The results indicate that the framework is self-consistent and operational across domains. The continuous side shows that richer realization detail still matters for strong late-stage re- finement, while the discrete side shows that the same metabolic controller can remain effective under a relatively lean route- based implementation. Overall, the study suggests that MMAO is best understood as a framework for generating adaptive search behavior from an endogenous resource economy rather than as a fixed operator set. Future work can proceed in at least three directions: stronger precision-oriented continuous instantiations, broader statisti- cally grounded empirical studies, and formal analysis of the budget dynamics that drive role and lifecycle adaptation. More broadly, the present paper aims to establish a clear and stable foundation on which such follow-up work can be carried out. REFERENCES [1] A. E. Eiben, R. Hinterding, and Z. 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