Extensible Modular Design

• Extensible modular design is a methodological paradigm that decomposes complex systems into weakly coupled, staged modules to ensure scalability and tractability.
• The approach employs local scenario discretization and decomposition algorithms such as Progressive Hedging and Benders to efficiently balance computational speed with optimality.
• It has broad applications in infrastructure, network, and multi-agent system design, demonstrating a favorable trade-off between solution quality and operational speed.

Extensible modular design is a methodological paradigm for enabling scalable, adaptive, and tractable optimization of complex systems composed of multiple interdependent subsystems, particularly under operational uncertainty and staged deployment. It emphasizes decomposing a large-scale problem into coordinated but only weakly coupled modules, leveraging localized uncertainty discretization and explicit inter-phase and inter-subsystem interfaces to maintain computational feasibility and design tractability, as exemplified in recent literature on infrastructure deployment, network design, and multi-agent system operation (Ho et al., 29 May 2025).

1. Foundational Framework: Stage-Based Modularization

Extensible modular design operates on the principle of staged decision-making, indexing system evolution by sequential phases $t=1,2,\dots,T$. In each phase, decisions $x^{t}$ are made based only on the realized uncertainties up to that stage, enforcing strong non-anticipativity. The decision vector for each phase partitions into subsystem-level variables (installation or design $y_i^t$, operational recourse $z_i^t(\xi^t)$). Crucially, design expansion is monotonic over time due to inter-phase carry-forward constraints: $x^{t}\geq x^{t-1},\qquad t=2,\dots,T$ This structure enables the representation of complex deployments, e.g., in infrastructure systems (water, energy) or multi-stage project rollout.

2. Mathematical Structure and Coupled Subsystem Interaction

The canonical mathematical program formulated under extensible modular design is a $T$-stage stochastic program with decision blocks for $n$ subsystems, integrated via stage-linking and subsystem-linking constraints: $$\min_{\{x^t\}_{t=1}^T} \sum_{t=1}^T \mathbb{E}_{\xi^t}\bigl[ c_t(x^t, \xi^t) \bigr]$$ Subject to:

• Inter-stage monotonicity: $x^t \geq x^{t-1}$
• Intra-stage coupling: system-level balance equations, e.g., $$A_{i} y_{i}^{t} + B_{i,j} z_{i}^{t}(\xi^{t}) + \dots \geq D^{t}(\xi^{t})$$
• Feasibility sets: $x^t \in X^t(\xi^t)$, encoding operational and structural constraints (often convex, e.g., linear or conic)

Subsystem interaction is explicitly quantified through sensitivity measures, such as the Jacobian norm: $$\alpha_{ij} := \Vert \frac{\partial g_{i,j}(x)}{\partial x_j}\Vert$$ with $\alpha_{ij}<\epsilon$ defining “weak” coupling eligible for approximate decoupling.

3. Local Scenario Discretization for Computational Scalability

A central innovation in extensible modular design is the local scenario discretization technique. Instead of global scenario trees (which scale as $O(|\Xi|^T)$), each subsystem $i$ and phase $t$ samples a compact, local scenario set $S_{t,i}$. The expected stage cost is approximated as

$$\mathbb{E}_{\xi^t}[c_t(x^t, \xi^t)] \approx \sum_{s\in S_t} p_{t,s}\;c_t(x^t, \xi^t_{s}),\qquad S_t = \bigcup_{i=1}^{n} S_{t,i}$$

Scenario generation leverages methods such as marginal quantization, moment-matching, or sensitivity-driven selection, maintaining statistical representativity within each module.

4. Decoupling of Weakly Interacting Subsystems

For systems with low inter-subsystem interaction, extensible modular design employs approximate decoupling. The full multi-subsystem problem is separated into parallel subproblems, each corresponding to a module, with weak couplings replaced by dual variables (Lagrange multipliers):

1. Identify weak interactions $(i,j)$ via $\alpha_{ij}<\epsilon$.
2. Drop or linearize weak coupling constraints.
3. Solve each subproblem independently (holding multipliers fixed).
4. Adjust multipliers via gradient or dual-ascent to restore consistency.

This block decomposition enables highly parallelizable solution schemes and drastically reduces the global dimensionality of the master program.

5. Decomposition Algorithms and Scalability Analysis

Two standard decomposition algorithms leveraged are:

• Progressive Hedging (PH): Each local scenario-subproblem solves independently per iteration, followed by scenario average updates. Non-anticipativity across scenarios is enforced via quadratic penalties ($\rho/2\|x^t_{s} - \bar x^t\|^2$).
• Benders (L-shaped) Decomposition: Design variables are decided in a master problem, with operational “recourse” problems per local scenario giving rise to feasibility and cost “cuts”.

Complexity comparison:

Solution Method	Complexity	Coupling
Global scenario tree	$O(|\Xi|^T)$	Full
Local discretization + split	$O(T n |S_{t,i}|)$	Parallelizable

Empirical results: Case studies demonstrate reduction in solution times from hours (full coupled) to minutes (local + decoupled), with small optimality gaps (e.g., $+2.5\%$ cost increase with $25\times$ speedup in (Ho et al., 29 May 2025)).

6. Real-World Applications and Comparative Results

Infrastructures such as water-electricity networks, power grids, and food supply chains exemplify the use of extensible modular design. The water–electricity case paper in (Ho et al., 29 May 2025) demonstrates:

Method	Computation Time	Expected Cost (\$M)	Flexibility (150% peak)	Robustness (worst-case)
Fully coupled	240 min	100.0	99%	No unmet demand
Local discretization only	65 min	101.2 (+1.2%)	97%	5% shortage, 2 days
Local + decoupling (modular)	10 min	102.5 (+2.5%)	95%	8% shortage, 3 days

This suggests that modular extensibility achieves an effective compromise between tractability and solution quality, especially for systems with many weak interactions.

7. Adaptability to Other Domains and Generalization

The modular extensibility paradigm has broad applicability:

• Regional water: Decouples reservoirs vs. treatment, accounting for local rainfall/demand.
• Power grid expansion: Parallelizes transmission, generation, storage.
• Supply chains: Modularizes farm, processing, distribution.

Adopted workflow:

1. Identify deployment stages.
2. Define subsystem-specific uncertainties.
3. Assign local scenario sets.
4. Quantify and threshold subsystem couplings.
5. Apply decomposition (PH/Benders) per module.

A plausible implication is that as system scale ($n$, $T$) grows, extensible modularity with local scenario management is the only computationally feasible path for real-time or repeated design-optimization cycles.

8. Limitations and Trade-offs

Approximate modularization is optimal only when subsystem interactions truly are weak; excessive decoupling leads to greater cost and reduced system robustness under severe stress scenarios. The methodology is sensitive to the tuning of scenario selection, coupling thresholds, and dual multiplier updates. Direct application to domains with strong nonlocal couplings requires either retaining full joint optimization or hybridizing with tighter scenario coordination.

Conclusion

Extensible modular design, as formalized in stage-based, locally discretized, and weakly coupled optimization frameworks (Ho et al., 29 May 2025), offers a mathematically rigorous and practically scalable mechanism to design, deploy, and adapt large-scale engineered systems under uncertainty. It achieves this by modularizing temporally and structurally, managing uncertainty via local scenarios, and exploiting algorithmic decomposability for high computational efficiency with marginal performance loss. This architectural paradigm is rapidly gaining traction across infrastructure planning, network design, and complex multi-agent system management, providing a robust template for managing system evolution in operationally uncertain environments.