Logic-Based Benders Decomposition (LBBD)

• Logic-Based Benders Decomposition (LBBD) is a framework that divides large-scale optimization problems into a master problem and one or more subproblems, leveraging logical inference to generate effective cuts.
• It employs diverse cut strategies such as strengthened nogood cuts and analytic logic-based cuts to iteratively refine solutions and ensure convergence in mixed discrete and continuous settings.
• LBBD is widely applied in scheduling, vehicle routing, robust optimization, and simulation-driven decision making, offering significant computational efficiency and scalability.

Logic-Based Benders Decomposition (LBBD) is an advanced framework for partitioning large-scale or structurally complex optimization problems into a master problem and one or more subproblems. Unlike classical Benders decomposition—which requires the subproblem to be solved as a linear (or convex) optimization—LBBD generalizes the approach so that the subproblem may be a combinatorial, nonconvex, or otherwise arbitrary optimization or feasibility check. The core principle is to exploit the logical, structural, or combinatorial properties of the subproblem to derive Benders cuts, allowing for effective solution of large problems that blend discrete and continuous domains or that couple complex scheduling and resource allocation with global task constraints.

1. Principles and Mathematical Framework of LBBD

LBBD begins with a problem of the general form: $$\min_{x, y} \quad f(x, y) \quad \text{subject to} \quad C(x, y), \quad C'(x), \quad x \in D_x, \quad y \in D_y,$$ where $x$ are master variables and $y$ are subproblem variables. By fixing $x$ at a candidate solution $x_k$, the subproblem becomes: $$\min_{y \in D_y} \quad f(x_k, y) \quad \text{subject to} \quad C(x_k, y).$$ Instead of requiring $C(x_k, y)$ to yield an LP or NLP, LBBD exploits inference duals—logical or constraint-programming (CP) based deductions—to prove lower bounds or establish feasibility.

Benders cuts in LBBD are logic-based constraints of the form: $z \geq B_x(x)$ where $B_x(x)$ is a function of the master variables capturing the outcome of the subproblem inference dual for the fixed master solution $x$.

In practice, the iterative LBBD algorithm alternates between:

• Solving the master problem subject to accumulated Benders cuts.
• Solving subproblems for candidate master solutions and generating new logic-based cuts if necessary.

This generality enables LBBD to address problems with subproblems that are themselves integer, combinatorial, or constraint programming models, bypassing the limitations of LP duality.

2. Cut Generation and Strengthening Strategies

The effectiveness of LBBD hinges on the quality and informativeness of the generated cuts. The following strategies are prominent:

a) Strengthened Nogood Cuts: After identifying an assignment that yields suboptimal, infeasible, or costly subproblem solutions, specialized cuts (nogoods) are constructed to preclude the repetition of these assignments. For example, in scheduling,

$$M_i \geq M^* \quad \text{when} \ \prod_{j \in J'} x_{ij} = 1,$$

where $J'$ is a critical subset of jobs whose scheduling causes a long makespan $M^*$. These are then linearized for master problem integration (Hooker, 2019).

b) Analytic Logic-Based Cuts: When the subproblem exhibits special structure, analytic cuts exploit domain knowledge for even tighter bounds. For instance, in minimum-makespan scheduling, cuts of the form

$$M_i \geq M^* - \sum_{j\in J'} P_{ij} (1 - x_{ij}) + (\max_{j\in J'}{d_j} - \min_{j\in J'}{d_j})$$

can be used to communicate subproblem tightness directly to the master (Elci et al., 2020).

c) Feasibility and Optimality Cuts by Inference: If the subproblem is infeasible for a master solution $x$, a logic-based "no-good" or indicator cut is added; if it is feasible, an optimality cut (possibly with inferred lower bounds) is derived.

d) Use of Partial Decomposition and Relaxations: Sometimes LP relaxations or partial decompositions by route or task are incorporated to provide early, informative, and easy-to-generate cuts (as in bus fleet electrification (Legault et al., 7 Aug 2025)).

e) Hybrid Enhancements: Benders and LBBD are often augmented with specialized heuristics (RENS), column generation, and neighborhood search techniques to accelerate convergence and improve primal solution quality (e.g., (Daryalal et al., 2021, Avgerinos et al., 4 May 2025)).

3. LBBD Algorithmic Variants and Integration with Solvers

The classic LBBD approach (iterative solve-and-cut) is complemented by more advanced variants:

Branch and Check: Rather than solving the master to completion between each cut, this method explores the master’s search tree with branch-and-bound, checking subproblem feasibility or optimality at each integer node and adding cuts dynamically (Hooker, 2019). This approach is computationally advantageous when the master problem is challenging or the subproblem solution is fast.

Integration with Diverse Solvers: LBBD is natively compatible with MIP, constraint programming, and even simulation. For instance, simulation-based subproblems yield cuts based on observed (possibly stochastic) performance measures, which are then imposed on the master problem (Forbes et al., 2021).

Cut Selection and Management: Advanced methods select among multiple candidate cuts (e.g., "deepest" or "closest" cuts in Magnanti–Wong style) to maximize the impact per iteration, as well as sparser, monotone, or indicator cuts that exploit the structure of the value function (e.g., in fleet electrification (Legault et al., 7 Aug 2025)).

4. Domains of Application and Representative Models

LBBD has demonstrated efficacy across a broad spectrum of large-scale and hybrid combinatorial optimization settings, including:

• Scheduling and Assignment: Applications to job assignment, makepsan minimization, resource-constrained parallel machine problems, and health care scheduling (with CP-based subproblems) yield substantial computational gains over monolithic MIP or CP (Elci et al., 2020, Avgerinos et al., 2023, Cappanera et al., 2023).
• Vehicle Routing and Logistics: LBBD is effective in complex vehicle routing with synchronization, intermodal transportation, and pickup/delivery with transfers, where it partitions path selection (master) from resource/time-synchronization constraints (subproblem) (Daryalal et al., 2021, Avgerinos et al., 2022, Avgerinos et al., 4 May 2025).
• Robust and Stochastic Optimization: LBBD frameworks accommodate robust min-max regret (Assunção et al., 2020), stochastic distributed operating room scheduling (Guo et al., 2019), K-adaptability (Ghahtarani et al., 2022), and scenario-based probabilistic set covering (Liang et al., 24 Jan 2025).
• Network Planning and Infrastructure: Large-scale network migration, fleet electrification, and broadcasting frequency plans have benefited from LBBD due to their natural decomposability and complex subproblem logical structure (Daryalal et al., 2021, Legault et al., 7 Aug 2025).
• Logic and Simulation Integration: LBBD directly mediates between simulation-based performance evaluation for resource allocation and assignment decisions, allowing non-analytical outcomes to guide optimization (Forbes et al., 2021).

A summary table of diverse domains and their subproblem types:

Domain	Master Problem	Subproblem
Job Scheduling	Assignment/MIP	CP/Analytical (makespan, tardiness, etc.)
Vehicle Routing/Network Migr.	Routing decisions/MIP	CP+Column Gen. (routing, time windows)
Robust Optimization	Discrete decisions/MIP	NP-hard subproblems (min, min-max regret)
Electrification Planning	Investment decisions	Integer scheduling with logical monotonicity
Simulation-based Allocation	Allocation/MIP	Simulation oracle with monotonic cuts

5. Algorithmic Enhancements, Scalability, and Theoretical Guarantees

Several enhancement strategies recurrently appear in state-of-the-art applications:

• Efficient Cut Separation: Polynomial-time or column-oriented algorithms are developed for separating feasibility or optimality cuts, particularly in stochastic or robust settings where scenario explosion is a risk (Liang et al., 24 Jan 2025).
• Hybrid Metaheuristics: LBBD is warm-started or hybridized with large-neighborhood search (LNS), adaptive heuristics, or randomized neighborhood resampling to improve scalability for very large instances (Avgerinos et al., 4 May 2025).
• Convergence and Optimality: For 0-1 min-max regret and combinatorial robust optimization, it is proved that the LBBD algorithm converges in finitely many steps by ensuring each iteration introduces a unique cut and leveraging the finiteness of the feasible subproblem space (Assunção et al., 2020, Ghahtarani et al., 2022).
• Exactness Across Diverse Subproblem Types: The approach is rigorously shown to yield globally optimal solutions, independent of whether subproblems are linear, integer, or logic-based, under mild conditions requiring only well-posedness and finiteness (Hooker, 2019).

6. Impact, Challenges, and Extensions

LBBD provides a paradigm for integrating multiple solution technologies (MIP, CP, simulation, heuristic) within a cohesive decomposition framework. Its flexibility comes at the cost of problem-specific cut design, which requires deep understanding of subproblem inference structure—contrasting with the fully generic nature of LP-based Benders.

The literature documents numerous extensions:

• Hybrid Quantum–Classical LBBD: Benders decomposition assisted by quantum processors (e.g., neutral atom QPUs) via MILP-to-QUBO transformation for the master problem, enabling exploration of large binary solution spaces (Naghmouchi et al., 8 Feb 2024).
• Stochastic Programming with Integer/CP Recourse: LBBD outperforms the integer L-shaped method in two-stage stochastic scheduling, especially when second-stage problems lack strong LP relaxations (Elci et al., 2020, Guo et al., 2019).
• Nonlinear and Hybrid Systems: Integration with signal temporal logic in robotic motion planning, where master problems handle high-level discrete planning and subproblems enforce nonlinear, continuous, and logical constraints (Ren et al., 18 Aug 2025).

A key challenge remains in generalizing cut generation for arbitrary subproblem structures while maintaining computational efficiency and convergence guarantees.

7. Survey of Proven Benefits and Practical Implications

Across a range of documented applications, LBBD achieves:

• Orders-of-magnitude reductions in total variables compared to monolithic MIP, particularly when integer decisions can be isolated in the master (Hanbazazah et al., 2018, Daryalal et al., 2021).
• Superior solution times and tighter optimality gaps on large-scale stochastic and robust problems compared to direct MIP or classical Benders alternatives, e.g., closing gaps below 1–5% for up to 500 rows × 5,000 columns × 2,000 scenarios in probabilistic set covering (Liang et al., 24 Jan 2025), or producing optimal and near-optimal solutions for electrification and migration planning problems that were previously intractable (Legault et al., 7 Aug 2025).
• The ability to exactly and efficiently solve logic- and simulation-based problems (e.g., ASP for scheduling with no-good-based cuts, simulation-embedded allocation with monotonic cuts) that were previously considered only tractable for heuristics (Cappanera et al., 2023, Forbes et al., 2021).
• Novel convergence and cut-tightening properties in multi-objective and min-max-min robustness contexts, including provably finite convergence and theoretical tightness results in bi-objective Benders (Raith et al., 2022).

LBBD thus offers a theoretically resilient and practically scalable approach for tackling the combinatorial explosion and integration complexities of modern optimization problems, particularly as problem formulations grow to encompass logic, simulation, stochastic, and nonconvex constraints.