Relax-Round-Resolve Pipeline

• Relax-Round-Resolve Pipeline is a modular heuristic framework that relaxes, rounds, and resolves mixed-integer and nonconvex problems for high-quality feasible solutions.
• It employs continuous relaxations and tailored rounding techniques to handle challenging constraints in domains like power systems, water networks, and clustering.
• Variants such as relax–tighten–round–resolve enhance solution quality with bound tightening and cutting planes, improving optimality certificates and feasibility.

The relax–round–resolve pipeline is a heuristic algorithmic framework for solving mixed-integer and combinatorial optimization problems with challenging nonconvex or nonlinear constraints. It proceeds by first relaxing integrality or discrete restrictions to solve a tractable continuous surrogate, then applying a rounding procedure to restore discrete or integer feasibility, and finally resolving the continuous subproblem with discrete decisions fixed. This modular paradigm has found success in large-scale problems in power systems, water networks, and clustering, supporting both the computation of high-quality feasible solutions and upper/lower bounds on optimality. Variants include the relax–tighten–round–resolve pipeline, which includes explicit relaxation-strengthening through bound tightening or additional cutting planes.

1. Methodological Framework

The canonical relax–round–resolve pipeline can be described through three principal stages:

1. Relax: Form a continuous or convex (e.g., linear or semidefinite) relaxation of the original mixed-integer (or otherwise discrete) problem, typically by replacing $\{0,1\}$ constraints with $[0,1]$ intervals or, in the case of binary/constrained combinatorics, relaxing to the convex hull. For nonlinear or nonconvex problems, additional linearizations or outer approximations are introduced as necessary (Pecci et al., 2020, Gómez et al., 20 Jan 2025, Tong et al., 29 Jan 2026).
2. Round: Map the (typically fractional) relaxed solution into a feasible integer or discrete configuration. Rounding may proceed via nearest-neighbor projections, greedy selection of top-$k$ components, sorting by rescaled relaxed values, or more nuanced, combinatorially aware algorithms that preserve problem-specific feasibility such as ramping constraints in unit commitment or device grouping in power flows (Gómez et al., 20 Jan 2025, Pecci et al., 2020, Tong et al., 29 Jan 2026).
3. Resolve: With discrete variables fixed, solve the induced continuous (often nonconvex) subproblem—e.g., resolving AC power flows or nonlinear user quality in water network control—to obtain a performance- and constraint-feasible solution, possibly certifying bounds or gaps with respect to the relaxed optimum (Gómez et al., 20 Jan 2025, Pecci et al., 2020, Tong et al., 29 Jan 2026).

A generic pipeline is summarized in the following table:

Stage	Core Operation	Output
Relax	Continuous/convex relaxation (NLP, LP, SDP)	Fractional solution, lower bound
Round	Discrete-variable restoration (mapping/heur.)	Integer/discrete assignment
Resolve	Continuous subproblem with fixed discrete vars	Feasible solution, upper bound

Variants such as relax–tighten–round–resolve augment this process with an intermediate tighten step involving optimization-based bound tightening (OBBT) or cut generation to strengthen the relaxation (Pecci et al., 2020).

2. Pipeline Instantiations in Representative Domains

The relax–round–resolve methodology has been concretely instantiated in several domains:

• Clustering with Convex Relaxations: For $k$-median and $k$-means, relaxations are formed via LPs or SDPs. Under appropriate separation regimes on the underlying data (the "unit balls" model), the relaxation itself produces an integral solution—"round-free"—thus bypassing the explicit rounding step entirely. For example, the $k$-median LP is integral for any cluster separation $\Delta > 2+\varepsilon$; $k$-means SDP is integral for $\Delta > 2\sqrt2(1+\sqrt{1/m})$ (Awasthi et al., 2014).
• Mixed-Integer AC Optimal Power Flow (UC-ACOPF): The unit commitment problem with AC power flow (UC-ACOPF) is relaxed by removing binary constraints, then rounded using model-informed scaling (e.g., Re-RUC or Re-Power rescalings) and formulaic assignment (e.g., enhanced rounding procedures that explicitly enforce minimum up/down and ramping constraints), followed by solving the ACOPF with fixed commitments (Gómez et al., 20 Jan 2025).
• Volt/VAR Optimization (VVO) in Transmission Networks: For control of discrete OLTC and CB devices, an initial MINLP is relaxed to continuous device settings, rounded to nearest manufacturer-feasible settings, then the full AC power flow is solved with these fixed for operational feasibility (Tong et al., 29 Jan 2026).
• Water Networks (Valve/Booster Placement): A complex nonconvex MINLP describing pressure reducing valves and booster placements is linearized (relax), further tightened via OBBT (tighten), rounded (top-$n$ selection), and resolved in the original nonlinear physical space with fixed discrete decisions, yielding tight optimality gaps and high feasibility (Pecci et al., 2020).

3. Mathematical Underpinnings

The pipeline's efficacy critically depends on the tightness and tractability of the continuous relaxation, as well as the structure-preserving nature of the rounding and resolve procedures. In some problems—most notably clustering under large separation—the relaxation is provably integral, so rounding and resolve reduce to a trivial labeling pass. For generic MINLPs or discrete optimization settings:

• Relaxation Quality: Relaxations may be via LP, SDP, or more general nonlinear programming. For nonconvex quadratic/bilinear constraints, the pipeline relies on polyhedral envelopes or RLT-based bounds (Pecci et al., 2020). SDP relaxations may admit dual certificate construction (as in $k$-means) for integrality guarantees (Awasthi et al., 2014).
• Rounding Schemes: Rounding strategies are problem-specific: nearest-neighbor for continuous device settings (Tong et al., 29 Jan 2026), top-$k$-selection for sparse control (Pecci et al., 2020), or more sophisticated routines that integrate system-operational or combinatorial feasibility. Rescaling fractional variables before rounding (Re-RUC, Re-Power) can drastically improve feasibility, particularly for binary status in UC-ACOPF (Gómez et al., 20 Jan 2025).
• Resolution: The fixed-discrete-variable continuous subproblem is commonly a nonlinear program (NLP). Efficient NLP solvers (e.g., IPOPT, penalized SLP) are typically used, and practical convergence is often good even for large-scale transmission networks or water systems, though infeasibility may arise in rare cases (Tong et al., 29 Jan 2026).

4. Computational Properties and Performance

Extensive computational evidence demonstrates that the relax–round–resolve pipeline yields practically feasible and high-quality solutions across domains:

• Power Systems: In transmission VVO, the pipeline consistently achieves substantial voltage and VAR profile improvements (e.g., MAE_v reductions of 10–20%, and VAR reductions of 30–60%), with runtimes compatible with real-time operator needs (subminute to minutes on cases with up to 13,659 buses) (Tong et al., 29 Jan 2026). In UC-ACOPF, physics-informed rounding strategies produce feasible schedules with cost gaps under 1%, outperforming naïve rounding (Gómez et al., 20 Jan 2025).
• Water Networks: The relax–tighten–round pipeline for valve and booster station placement yields solutions with optimality gaps comparable to physical model uncertainty, and consistently outperforms global MINLP solvers in both solution quality and feasibility (Pecci et al., 2020).
• Clustering: When separations are sufficiently large, the relaxations themselves yield the ground-truth clustering, directly certifying optimality and bypassing manual rounding (Awasthi et al., 2014).

The table below summarizes major pipeline attributes by domain:

Domain	Relaxation Type	Rounding	Resolve	Key Empirical Outcome
VVO (Power)	NLP	Nearest-neigh.	NLP	10–20% voltage, 30–60% VAR MAE reduction
UC-ACOPF	NLP	Rescale+round	NLP	$\leq$1% cost gap, high feasibility
Water networks	LP (tightened)	Top-$n$ pick	NLP+MILP	Feasible, low gap solutions
Clustering	LP/SDP	Trivial (*)	None/labeling	Certifies ground-truth when separation large

(*) Rounding is vacuous under proven integrality of relaxed solution.

5. Algorithmic Extensions and Limitations

Several extensions and practical considerations are observed:

• Tightening Steps: In relax–tighten–round–resolve, OBBT and multiple linear cuts can considerably improve relaxation bounds and rounding quality, especially for non-convex quadratic/bilinear terms (Pecci et al., 2020).
• Iterative Approaches: Integrated frameworks such as the Feasibility Pump wrap the pipeline in an iterative improvement process, alternating between continuous relaxations (with penalization) and advanced rounding (with randomization or constraint-aware assignment), increasing the likelihood of finding feasible, low-cost solutions (Gómez et al., 20 Jan 2025).
• Optimality Guarantees: The pipeline is inherently heuristic when applied to general MINLPs; global optimality is not guaranteed except in provably integral regimes (as in $k$-median with large separation) (Awasthi et al., 2014). Empirically, however, bounds and gap certificates can often be computed efficiently.
• Failure Modes: Direct (naïve) rounding can yield infeasible or high-cost solutions, especially without rescaling or constraint-aware heuristics. Recovery strategies include local search/repair, randomized perturbations, or feasibility pumps (Gómez et al., 20 Jan 2025, Pecci et al., 2020).
• Scope of Applicability: The pipeline is broadly applicable to MINLPs whose only non-convexities are single-row quadratics or bilinears, or to combinatorial optimization problems where tight convex relaxations are tractable (Pecci et al., 2020).

6. Connections, Contrasts, and Future Directions

The relax–round–resolve approach is distinguished from purely iterative heuristics (e.g., Lloyd’s $k$-means), which lack global bounds and may fail even with large separation or favorable initialization (Awasthi et al., 2014). LP- and SDP-based relaxations, when tight, offer certificates of optimality and facilitate parameter-free operation. Open challenges include:

• Extending “no-rounding” integrality results to broader models, e.g., to overlapping balls or mixture-of-Gaussians for clustering (Awasthi et al., 2014).
• Combining relaxation-based pipelines with fast local refinement or metaheuristics to further improve solution feasibility and quality (Gómez et al., 20 Jan 2025).
• Developing adaptive and dual-informed rounding or grouping strategies to strengthen solution quality.
• Generalizing the framework to other combinatorial problems, such as graphical model inference or stable partitioning under perturbation (Awasthi et al., 2014).

A plausible implication is that as solver technology for NLP, MILP, and SDP continues to advance, the relax–round–resolve pipeline will become an increasingly central paradigm in the computational resolution of large-scale, nonconvex mixed-integer problems, bridging the gap between tractable relaxations and operational feasibility across domains.