Multi-level Building Function Optimization

• ML-BFO is a multilevel design pattern that decomposes complex building challenges into surrogate tasks (e.g., value functions, Pareto sets, labels) for enhanced scalability.
• It integrates methodologies from generalized Benders decomposition, sequential Pareto search in design, urban classification, and hierarchical predictive control.
• ML-BFO improves optimization efficiency by propagating level-specific cuts, Pareto sets, or resource allocations, ensuring near-optimal solutions with reduced computational effort.

Searching arXiv for the cited papers to ground the article. Multi-level Building Function Optimization (ML-BFO) denotes a family of multilevel optimization constructions rather than a single canonical algorithm. In the arXiv literature, the label is used in at least four technically distinct senses: as a generalized Benders/value-function methodology for multilevel and multistage mixed-integer linear optimization; as a sequential, multi-objective decomposition for building performance design; as a three-stage pipeline for building-level urban function classification from Points of Interest (POI) and footprints; and as a hierarchical optimization layer within a modular, physics-informed framework for building modeling and control (Bolusani et al., 2021, Talami et al., 2024, Chen et al., 10 Oct 2025, Jiang et al., 22 Jan 2026). Across these uses, the unifying motif is the replacement of a monolithic problem by level-wise surrogates, followed by upward propagation of cuts, Pareto sets, labels, or resource allocations.

1. Terminological scope and research lineages

The term ML-BFO appears in contexts that share multilevel structure but differ substantially in mathematical object, data model, and validation protocol. In the generalized decomposition setting, the “building function” is explicitly the value function of a lower-level optimization problem. In building performance design, ML-BFO is a staged sequential search in which design variables are partitioned into architecture and engineering levels, and only Pareto-optimal solutions are propagated forward. In urban classification, ML-BFO is a three-stage fusion pipeline that assigns one dominant function to each building by combining POI proximity, neighborhood autocorrelation, and landmark-informed correction. In BESTOpt, ML-BFO maps onto a hierarchy from cluster to component, with coordinated control and optimization across thermal, electrical, and water domains (Bolusani et al., 2021, Talami et al., 2024, Chen et al., 10 Oct 2025, Jiang et al., 22 Jan 2026).

Research line	Core object	Level-to-level propagation
Generalized Benders multilevel optimization	Nested value functions $V_i$	Optimality and feasibility cuts
Sequential building performance design	Pareto sets over staged design variables	Pareto-optimal solutions
Building function classification	Hard labels and category scores	Iterative refinement and buffer overrides
BESTOpt hierarchical control	Allocations, setpoints, and consensus variables	Downward dispatch and upward aggregation

This suggests that ML-BFO is best understood as a multilevel design pattern. Its concrete instantiation depends on whether the lower-level object is a value function, a Pareto frontier, a categorical label field, or a control subproblem.

2. Value functions, generalized Benders decomposition, and multilevel optimization

In the generalized Benders formulation, ML-BFO is the methodology of constructing, approximating, and optimizing the “building functions” associated with each level, where each building function is the value function of a lower-level problem. For a $k$-level problem with additively separable objective and linear constraints, the top level is written as

$\min_{x_1 \in X_1} f_1(x_1)+V_2(x_1),$

with nested definitions

$$V_2(x_1)=\min_{x_2 \in X_2(x_1)} f_2(x_1,x_2)+V_3(x_1,x_2),$$

and

$$V_i(\cdot)=\min_{x_i \in X_i(\cdot)} f_i(\cdot,x_i)+V_{i+1}(\cdot,x_i), \quad i=2,\ldots,k-1,$$

ending with $$V_k(\cdot)=\min_{x_k \in X_k(\cdot)} f_k(\cdot,x_k)$$. Auxiliary variables $\theta_i$ are introduced as master-level surrogates for $V_i$, so that lower-level variables are projected out and replaced by cuts that underestimate the corresponding value functions (Bolusani et al., 2021).

For LP subproblems, the canonical construction follows strong duality. If

$$V(x)=\min_y \{d^\top y:\; Ay \ge b-Bx,\; y \ge 0\},$$

then the dual is

$$V(x)=\max_{u \ge 0}\{u^\top(b-Bx):\; A^\top u \le d\}.$$

An optimal dual solution $k$0 at $k$1 yields the Benders optimality cut

$k$2

If the subproblem is infeasible, an extreme ray $k$3 yields the feasibility cut

$k$4

The multilevel extension nests these constructions. At level $k$5, the cut has the generic form

$k$6

and cut propagation proceeds recursively from level $k$7 upward. Multi-cut strategies may add one cut per scenario, right-hand side, extreme point, or extreme ray. The top-level master minimizes $k$8 subject to all accumulated optimality and feasibility cuts.

The principal complication arises when lower levels are MILPs rather than LPs. Classical convex duality no longer applies, so the method uses dual functions extracted from branch-and-bound trees. For a MILP value function $k$9, the branch-and-bound lower bounding function is

$\min_{x_1 \in X_1} f_1(x_1)+V_2(x_1),$0

where $\min_{x_1 \in X_1} f_1(x_1)+V_2(x_1),$1 are leaf nodes, $\min_{x_1 \in X_1} f_1(x_1)+V_2(x_1),$2 is a dual solution of the LP relaxation at node $\min_{x_1 \in X_1} f_1(x_1)+V_2(x_1),$3, and $\min_{x_1 \in X_1} f_1(x_1)+V_2(x_1),$4 is obtained from reduced costs and bounds. Because this is a min-of-affines, the resulting master representation is nonconvex and must be linearized with binary selectors and big-$\min_{x_1 \in X_1} f_1(x_1)+V_2(x_1),$5, or handled through multi-cut separation. For lexicographic bilevel MILPs, the construction uses a strong dual function $\min_{x_1 \in X_1} f_1(x_1)+V_2(x_1),$6 and then replaces the internal MILP value term $\min_{x_1 \in X_1} f_1(x_1)+V_2(x_1),$7 with a strong primal upper bound $\min_{x_1 \in X_1} f_1(x_1)+V_2(x_1),$8, producing a valid surrogate $\min_{x_1 \in X_1} f_1(x_1)+V_2(x_1),$9.

The framework contrasts directly with Fortuny–Amat KKT reformulations. For bilevel LPs, KKT plus big-$$V_2(x_1)=\min_{x_2 \in X_2(x_1)} f_2(x_1,x_2)+V_3(x_1,x_2),$$0 yields a single-level MILP, but the data explicitly notes the associated risks: weak relaxations, numerical issues, large complementarity systems, and a requirement for convexity and constraint qualification. ML-BFO avoids explicit KKT conditions and is particularly advantageous when lower levels are MILP or lexicographic MILP, where KKT is not directly applicable. Under polyhedral feasible regions, finite extreme points and rays, valid master linearizations, and finite leader projection, finite convergence follows when each added cut is strong at the current iterate (Bolusani et al., 2021).

3. Sequential multi-objective building performance design

In building performance design, ML-BFO refers to a staged decomposition of building functions into a two-level hierarchy that proved most effective: Level 1 is architecture, consisting of building geometry and building fabric; Level 2 is engineering, consisting of HVAC system and controls. The decision vector is partitioned as

$$V_2(x_1)=\min_{x_2 \in X_2(x_1)} f_2(x_1,x_2)+V_3(x_1,x_2),$$1

with $$V_2(x_1)=\min_{x_2 \in X_2(x_1)} f_2(x_1,x_2)+V_3(x_1,x_2),$$2 for geometry, $$V_2(x_1)=\min_{x_2 \in X_2(x_1)} f_2(x_1,x_2)+V_3(x_1,x_2),$$3 for fabric, $$V_2(x_1)=\min_{x_2 \in X_2(x_1)} f_2(x_1,x_2)+V_3(x_1,x_2),$$4 for HVAC, and $$V_2(x_1)=\min_{x_2 \in X_2(x_1)} f_2(x_1,x_2)+V_3(x_1,x_2),$$5 for controls. The paper adopts the bi-objective formulation

$$V_2(x_1)=\min_{x_2 \in X_2(x_1)} f_2(x_1,x_2)+V_3(x_1,x_2),$$6

where $$V_2(x_1)=\min_{x_2 \in X_2(x_1)} f_2(x_1,x_2)+V_3(x_1,x_2),$$7 is total space-heating energy demand during occupied periods including supply and distribution energy, and $$V_2(x_1)=\min_{x_2 \in X_2(x_1)} f_2(x_1,x_2)+V_3(x_1,x_2),$$8 is occupied discomfort hours based on PMV thresholds. Discomfort hours are counted when $$V_2(x_1)=\min_{x_2 \in X_2(x_1)} f_2(x_1,x_2)+V_3(x_1,x_2),$$9 or $$V_i(\cdot)=\min_{x_i \in X_i(\cdot)} f_i(\cdot,x_i)+V_{i+1}(\cdot,x_i), \quad i=2,\ldots,k-1,$$0 during occupied hours (Talami et al., 2024).

The sequential rule is that future stages are held at baseline values during each stage’s search: $$V_i(\cdot)=\min_{x_i \in X_i(\cdot)} f_i(\cdot,x_i)+V_{i+1}(\cdot,x_i), \quad i=2,\ldots,k-1,$$1 At level $$V_i(\cdot)=\min_{x_i \in X_i(\cdot)} f_i(\cdot,x_i)+V_{i+1}(\cdot,x_i), \quad i=2,\ldots,k-1,$$2, the within-level exhaustive search yields a Pareto set $$V_i(\cdot)=\min_{x_i \in X_i(\cdot)} f_i(\cdot,x_i)+V_{i+1}(\cdot,x_i), \quad i=2,\ldots,k-1,$$3, and only these non-dominated solutions are propagated to the next stage. The procedure is deterministic and exhaustive within each stage: enumerate all stage combinations, evaluate objectives, extract the Pareto set, and pass that set forward. An initial run starts from a single baseline point; an iterative run uses the previous run’s Pareto set as multiple starting points.

The study tested 24 configurations across three grouping strategies and four starting points, each in initial and iterative modes: no grouping, element-grouped, and field-grouped. The field-grouped, two-stage partition consistently recovered the global Pareto set obtained by full factorial search. On a multi-scale office-building case study with design spaces ranging from 874 to 1,036,800 options, and with boundary-condition variants reaching 4,147,200 options, the two-stage sequential process identified the same Pareto-optimal solutions as the full factorial search across all four scales and variations of problem formulations, demonstrating 100% effectiveness and reliability. The best-performing field-grouped iterative configuration required up to 100,700 function evaluations for the large-scale case, corresponding to approximately 8.8% of full factorial load and a reported 91.2% reduction in computational effort. Under the same evaluation budget, NSGA-II achieved only 73.5% of the global optima, while ML-BFO matched the full factorial’s energy and comfort values with mean percentage difference approaching 0% (Talami et al., 2024).

The reported interpretation is strongly hierarchical. Architectural decisions shape demand profiles, while engineering decisions adapt to those profiles. This suggests that ML-BFO performs well when partitioning respects dominant physical interactions and when Pareto propagation prunes combinatorial growth without discarding the critical trade-off structure.

4. Building-level urban function classification

A different use of ML-BFO appears in large-scale urban analytics, where it denotes a three-stage method that fuses Baidu POI records with building footprints to produce fine-grained building-level functional labels across the Guangdong–Hong Kong–Macao Greater Bay Area. The dataset assigns one of five functional categories to nearly four million buildings across six core cities: Guangzhou, Shenzhen, Hong Kong, Zhuhai, Zhongshan, and Foshan. The unified 5-class scheme comprises Residential, Commercial Services, Public Services, Technology and Industry, and Educational and Cultural. Mixed-use is not modeled explicitly; a dominant-function strategy is used (Chen et al., 10 Oct 2025).

Stage 1 generates candidate labels through spatial overlay with proximity weighting. For building $$V_i(\cdot)=\min_{x_i \in X_i(\cdot)} f_i(\cdot,x_i)+V_{i+1}(\cdot,x_i), \quad i=2,\ldots,k-1,$$4 and associated POIs $$V_i(\cdot)=\min_{x_i \in X_i(\cdot)} f_i(\cdot,x_i)+V_{i+1}(\cdot,x_i), \quad i=2,\ldots,k-1,$$5, the distance-decay score is

$$V_i(\cdot)=\min_{x_i \in X_i(\cdot)} f_i(\cdot,x_i)+V_{i+1}(\cdot,x_i), \quad i=2,\ldots,k-1,$$6

and local category de-biasing is introduced through

$$V_i(\cdot)=\min_{x_i \in X_i(\cdot)} f_i(\cdot,x_i)+V_{i+1}(\cdot,x_i), \quad i=2,\ldots,k-1,$$7

The per-building category score is then

$$V_i(\cdot)=\min_{x_i \in X_i(\cdot)} f_i(\cdot,x_i)+V_{i+1}(\cdot,x_i), \quad i=2,\ldots,k-1,$$8

and the hard candidate label is

$$V_i(\cdot)=\min_{x_i \in X_i(\cdot)} f_i(\cdot,x_i)+V_{i+1}(\cdot,x_i), \quad i=2,\ldots,k-1,$$9

Stage 2 performs iterative refinement using neighborhood label autocorrelation. For the $$V_k(\cdot)=\min_{x_k \in X_k(\cdot)} f_k(\cdot,x_k)$$0-nearest-neighbor graph, the update rule applies a strict-majority vote: $$V_k(\cdot)=\min_{x_k \in X_k(\cdot)} f_k(\cdot,x_k)$$1 This is intended to damp isolated mislabels and promote spatial coherence without over-smoothing ambiguous borders.

Stage 3 applies function-related correction informed by High-level POI buffers. If a building intersects the buffer of a landmark POI of category $$V_k(\cdot)=\min_{x_k \in X_k(\cdot)} f_k(\cdot,x_k)$$2, its final label is overridden to $$V_k(\cdot)=\min_{x_k \in X_k(\cdot)} f_k(\cdot,x_k)$$3: $$V_k(\cdot)=\min_{x_k \in X_k(\cdot)} f_k(\cdot,x_k)$$4 High-level POIs include large shopping centers, major residential compounds, higher education campuses, large hospitals and government complexes, and sizable industrial or tech parks.

Because fine-grained building-level ground truth is scarce, the paper introduces the Building Function Matching Index (BFMI). Let $$V_k(\cdot)=\min_{x_k \in X_k(\cdot)} f_k(\cdot,x_k)$$5 be the building-level probability vector derived from POI KDE heatmaps, and $$V_k(\cdot)=\min_{x_k \in X_k(\cdot)} f_k(\cdot,x_k)$$6 the predicted label encoding. Then

$$V_k(\cdot)=\min_{x_k \in X_k(\cdot)} f_k(\cdot,x_k)$$7

with $$V_k(\cdot)=\min_{x_k \in X_k(\cdot)} f_k(\cdot,x_k)$$8 by default. GBA-UBF reports a BFMI of approximately 0.58, markedly higher than the EULUC-China 2.0 baseline, and qualitative and field validation are reported to confirm semantic reliability and practical interpretability (Chen et al., 10 Oct 2025).

5. Hierarchical optimization and control in BESTOpt

Within BESTOpt, ML-BFO is mapped onto a cluster–domain–system/building–component hierarchy. The cluster level coordinates multiple buildings and systems; the domain level manages thermal, electrical, and water flows; the system/building level governs supplier systems and demander buildings; and the component level contains atomic devices such as fans, coils, pumps, chillers, heat pumps, ice tanks, PV, batteries, and EVs. All modules share a standardized state–action–disturbance–observation typology, and the framework supports centralized, decentralized, and hybrid or distributed control (Jiang et al., 22 Jan 2026).

At cluster level, a representative objective is

$$V_k(\cdot)=\min_{x_k \in X_k(\cdot)} f_k(\cdot,x_k)$$9

subject to power-balance and import/export constraints such as

$\theta_i$0

At building level, the framework uses local objectives of the form

$\theta_i$1

with thermal discomfort penalties based on zone-temperature bands. Component-level subproblems track system references while enforcing device physics, for example battery state-of-charge dynamics or fan and pump power models.

The distinctive feature is the embedding of physics priors into data-driven modules. BESTOpt’s PI-ModNN thermal dynamics use modular neural blocks for external heat transfer, internal gains, HVAC thermal load, and adjacent heat transfer, regularized by a physics-consistency term $\theta_i$2 in a total loss $\theta_i$3. Under an unseen condition in which HVAC was intentionally turned off, PI-ModNN achieved MAE $\theta_i$4 and MAPE $\theta_i$5, whereas LSTM yielded MAE $\theta_i$6 and MAPE $\theta_i$7. For occupancy forecasting, the paper reports one-day-ahead MAE $\theta_i$8 and RMSE $\theta_i$9. In a five-house cluster, PI-ModNN training time was reported as under five minutes per building using three months of data, and the case study demonstrated multi-building simulation and comparison under common control abstractions (Jiang et al., 22 Jan 2026).

The distributed formulation is consensus-based. An augmented Lagrangian enforces agreement between local building electrical imports and cluster-level grid variables, and alternating updates provide an ADMM-like decomposition. This is structurally consistent with the broader ML-BFO pattern: local subproblems remain physically expressive, while upper-level coordination handles shared constraints and global objectives.

6. Shared principles, limitations, and points of comparison

Across these formulations, ML-BFO is characterized by decomposition plus selective propagation. In generalized Benders methods, the propagated object is a lower-bounding cut; in sequential design optimization, it is a Pareto set; in urban classification, it is a refined label state; in BESTOpt, it is a coordinated allocation or setpoint. A plausible implication is that the term “building function” has two different roles in the literature: in mathematical programming it denotes a value function, while in urban analytics it denotes semantic building use.

The literature also makes clear that performance depends critically on how levels are defined. In generalized Benders settings, weak cuts, dual degeneracy, and big-$V_i$0 linearizations can slow convergence or damage numerical stability. In sequential building design, ungrouped and element-grouped variants deteriorated with increasing scale, while the field-grouped partition was crucial to 100% recovery of the full-factorial Pareto set. In urban classification, neighborhood size $V_i$1, category de-bias exponent $V_i$2, and buffer radii $V_i$3 are all consequential but not fully disclosed; the method also assumes a single dominant function per building and is exposed to POI timeliness issues, with about 8% label drift noted in the limitations. In BESTOpt, nonconvex device physics, mixed-integer actuation, and forecast error remain practical constraints on controller scalability and robustness (Bolusani et al., 2021, Talami et al., 2024, Chen et al., 10 Oct 2025, Jiang et al., 22 Jan 2026).

A common misconception would be to treat ML-BFO as a single universally defined algorithm. The arXiv record instead supports a narrower and more technical conclusion: ML-BFO is a reusable multilevel strategy whose concrete form is determined by the intermediate object being optimized or propagated. Where the lower-level object is convex or admits strong surrogate bounds, ML-BFO takes the form of cut-based value-function approximation. Where the problem is simulation-heavy and combinatorial, it becomes deterministic staged search with Pareto pruning. Where the target is semantic labeling over space, it becomes staged evidence aggregation and correction. Where the target is cyber-physical coordination, it becomes hierarchical predictive control with physics-informed local models.