Rolling Optimization Approach

• Rolling optimization is a dynamic decision-making framework that divides extensive, uncertain problems into overlapping subproblems solved with updated information.
• It integrates forecasts and real-time data within a receding horizon strategy to optimize multi-stage objectives efficiently under uncertainty.
• Key applications span energy systems, robotics, and resource management, balancing decision quality and computational tractability through window sizing techniques.

A rolling optimization approach refers to a broad class of decision-making frameworks in which a complex, dynamic, and often high-dimensional problem is decomposed into a sequence of overlapping, shorter-horizon subproblems, which are repeatedly solved as new information becomes available. This paradigm is prevalent in control, planning, resource allocation, combinatorial optimization, stochastic programming, robotics, energy systems, and numerous other fields. Each "roll" involves planning into the future (the look-ahead or prediction window), executing only the earliest decisions, updating the state with new observations or realizations, and then re-solving the problem on the shifted window. Rolling optimization enables adaptivity to uncertainty, tractability for long-horizon problems, and integration of real-time data.

1. Core Principles and Mathematical Structures

At its most general, rolling optimization is a receding-horizon or moving-horizon strategy. Denote the system state at time $t$ as $S_t$, controls or decisions as $x_t$, and uncertain exogenous data as $W_t$. At each roll, the approach:

• Precisely models and predicts system evolution over a finite look-ahead window $[t, t+H]$, constructing an optimization problem (typically a MILP, QCP, or constrained nonlinear program) using forecasts or scenarios for $W$ in that window.
• Solves for a sequence of decisions $x_{t:t+H}$ that optimize a suitable multi-stage objective (cost, loss, or expected reward), subject to system dynamics, operational constraints, and possibly uncertainty sets or scenario trees.
• Executes only the first-step decision $x_t$ (receding horizon), observes the realized $W_{t+1}$, updates the system state $S_{t+1}$, and shifts the window.
• Repeats until the planning interval is exhausted.

Rolling optimization may be viewed as an instance of closed-loop feedback control or as a policy search in stochastic control/Markov Decision Process (MDP) settings.

A stylized mathematical form is: $$\min_{x_{t:t+H}} \quad \sum_{j=0}^H \ell(S_{t+j}, x_{t+j})$$ subject to

$$S_{t+j+1} = f(S_{t+j}, x_{t+j}, W_{t+j+1}), \quad g(S_{t+j}, x_{t+j}) \leq 0,$$

with decision variables $x_{t:t+H}$ and uncertainty in $W$ handled via forecasts, scenario generation, robustification, or parametric "buffers" (III et al., 2017).

2. Rolling Optimization in Stochastic and Deterministic Settings

Rolling optimization serves both as a solution heuristic for high-dimensional stochastic control/optimization (when exact multistage dynamic programming is infeasible) and as a practically tractable approach for deterministic but computationally expensive resource planning.

• Stochastic Programming and Parametric Cost Function Approximation (CFA): In stochastic settings, the rolling horizon may solve a deterministic program with parametric adjustments (buffer stocks, forecast “fudges”) at each roll, then use simulation-based policy gradients to optimize the parameters $\theta$ for performance in the "base" stochastic model. This is formalized as

$$X_t(S_t|\theta) = \arg\min_{x_{tt},…,x_{t, t+H}} \sum_{t'=t}^{t+H} c_{tt'}^T x_{tt'} \text{ subject to } A_{tt'} x_{tt'} \leq b_{tt'}(\theta)$$

where $\theta$ are tuned by stochastic search or gradients to maximize expected performance $F(\theta)$ in the true system (III et al., 2017). Deterministic rolling approaches in industry are thus seen as a "parametric CFA policy search."

• Rolling Horizon with Full Stochastic or Robust Models: Modern frameworks may solve a two-stage or multi-stage stochastic program or a robust counterpart (e.g., robust MILP) at each roll, using scenario trees, uncertainty sets, or point forecasts, with only the first-stage actions implemented before re-optimization (Schlenkrich et al., 22 Feb 2024, Hönen et al., 2023, Spinelli et al., 23 May 2024).
• Approximate Dynamic Programming and Look-Ahead Policies: In dynamic programming, rolling optimization forms the basis for look-ahead policies and rollout strategies, such as the $h$-step rollout in Bayesian optimization, where $h$-step integrals are estimated to trade off exploration and exploitation (Lee et al., 2020).

3. Key Computational Techniques and Trade-offs

Variance Reduction and Computational Scaling

High-dimensionality and non-myopic objectives in rolling optimization often yield intractable integration or scenario enumeration. Sophisticated techniques include:

• Variance reduction in non-myopic acquisition function evaluation: For $h$-step lookahead in Bayesian optimization, estimation of $h$-dimensional integrals is made feasible via quasi-Monte Carlo (QMC), common random numbers (CRN) for smoother optimization landscapes across candidate decisions, and control variates for variance reduction. This enables practical rollout acquisition function computations otherwise intractable for large $h$ (Lee et al., 2020).
• Window Size and Tractability: The look-ahead or rolling window length $H$ controls the trade-off between decision quality and computational tractability. Larger $H$ leads to tighter bounds and more global optimization, but at significant cost and increased forecast/model uncertainty. Subproblems are commonly designed so each is solvable in real time (e.g., in seconds for energy markets (Cortés et al., 19 Mar 2025, Jiang et al., 13 Feb 2025), <2s per window for network verification (Zhao et al., 10 Jan 2024), hundreds of milliseconds for collaborative caching (Yang et al., 2019)).

Policy Search and Variable Fixation

• Policy Search in Acquisition Optimization: When direct optimization over a complex $h$-step rollout objective is costly, rolling optimization can be combined with discrete policy search over a set of myopic or parametric base policies, estimating the h-step reward for each candidate and adopting the one with the highest expected improvement. This reduces the complexity from global functional optimization to a portfolio search and is both efficient and robust (Lee et al., 2020).
• Learning-guided variable fixation: In large-scale combinatorial rolling optimization (e.g., job-shop scheduling), learning-based mechanisms can fix variables in overlapping subproblems that are highly likely to remain unchanged, reducing redundant computation and improving subproblem solve times and solution quality, as shown in L-RHO for Flexible Job-Shop Scheduling (Li et al., 18 Feb 2025).

4. Applications Across Domains

Rolling optimization is foundational in a wide array of scientific and engineering applications.

Application Area	Rolling Optimization Role	Example arXiv IDs
Bayesian optimization	Non-myopic acquisition function, rollout policy search	(Lee et al., 2020)
Stochastic resource allocation	Parametric CFA, buffer/forecast tuning with gradient-based search	(III et al., 2017)
Production and operations planning	Rolling lot-sizing under demand forecast evolution, scenario-based	(Schlenkrich et al., 22 Feb 2024)
Power and energy systems	Real-time dispatch, energy management, restoration, network reconfig	(Yao et al., 2019, Cortés et al., 19 Mar 2025, Hönen et al., 2023, Jiang et al., 13 Feb 2025, Feng et al., 2018)
Controller tuning	Data-driven PID re-tuning in a moving-horizon loop	(zhou, 2016)
Robotics, manipulation, and locomotion	Rolling path and whole-body trajectory optimization, model predictive control in real time for legged and wheeled robots	(Bjelonic et al., 2019, Bjelonic et al., 2018, Boroji et al., 11 Oct 2024, Yang et al., 23 Aug 2024, Putkaradze et al., 2017)
Neural network verification	Rolling-horizon decomposition/MIP subproblems for bound tightening	(Zhao et al., 10 Jan 2024)
Combinatorial optimization	Rolling-horizon variable fixation via learned policy for long FJSP	(Li et al., 18 Feb 2025)

In all these domains, rolling optimization frameworks enable adaptive response to uncertainty, computational scalability, and closed-loop integration of new information.

5. Performance Analysis and Limitations

Rolling optimization approaches exhibit distinct strengths and limitations:

• Adaptivity and Robustness: Frequent re-optimization provides resilience to model misspecification, exogenous uncertainty, and system changes—critical in disaster recovery, microgrid dispatch, and manufacturing (Feng et al., 2018, Cortés et al., 19 Mar 2025, Hönen et al., 2023).
• Solution Quality: Empirical studies demonstrate that rolling horizon optimization yields near-optimal performance compared to full-horizon or static schemes, with substantially lower computational requirements—often achieving $>$80% reductions in cost or time relative to greedy or static approaches (Yang et al., 2019, Schlenkrich et al., 22 Feb 2024, Spinelli et al., 23 May 2024, Li et al., 18 Feb 2025).
• Trade-offs: Excessive lookahead windows (large $H$) can lead to over-exploration or compounding errors under model uncertainty, and forecast errors accumulate, potentially degrading performance (Lee et al., 2020, III et al., 2017).
• Computational Complexity: MILPs, QCPs, and scenario-based models become infeasible for large $H$, high-dimensional spaces, and extensive uncertainty sets. Rolling horizon decomposes these into tractable pieces but requires window size tuning, warm-starting, and heuristic or learning-based acceleration to reach real-time performance (Zhao et al., 10 Jan 2024, Li et al., 18 Feb 2025, Cortés et al., 19 Mar 2025).
• Parameter and Policy Optimization: The effectiveness of parametric adjustment (e.g., buffer stock, reserve slack) hinges upon the expressivity of the parametric family and stochastic-gradient-based tuning (III et al., 2017).

6. Advanced Structuring: Decentralization, Parallelization, and Learning

• Multi-Agent Decentralization: Rolling optimization is amenable to decentralized, agent-based implementations—crucial in power grid restoration and data center resource allocation—with consensus protocols to coordinate local rolling-optimal schedules and maintain global feasible actions in dynamically changing networks (Feng et al., 2018).
• Parallel and Layered Decomposition: For large, layered networks (e.g., DNN verification), rolling-horizon bound propagation enables efficient, parallelizable MIP subproblem solutions, iteratively tightening performance bounds with high throughput (Zhao et al., 10 Jan 2024).
• Integration of Machine Learning: Recent approaches leverage machine learning to predict solution-invariant variables, prune search spaces, or inform policy selection within rolling horizon frameworks, yielding significant acceleration and robustness across problem types (Li et al., 18 Feb 2025, Lee et al., 2020).

7. Representative Limitations and Practical Guidelines

• Modeling Uncertainty: Rolling horizon can handle exogenous uncertainty through robustification, stochastic programming, or scenario sampling, but only as accurately as the underlying forecasts or distributional assumptions.
• No Guarantees of Global Optimality: Unless all system, forecast, and scenario models are exact, rolling optimization generally lacks global optimality—even in expectation—relative to an ideal dynamic programming policy.
• Window and Parameter Selection: Both window size and choice of parameterization (for buffer/slack adjustments) impact solution quality and computational load. Empirically, modest horizon lengths or parameter adjustments (rather than long lookahead) are best under substantial uncertainty or model error (Lee et al., 2020, III et al., 2017).
• Scenario and Forecast Management: Overly aggressive scenario or forecast expansion renders subproblems intractable; practical implementations enforce tight scenario selection, window management, and early termination rules (Yang et al., 2019, Zhao et al., 10 Jan 2024).

• Efficient Rollout Strategies for Bayesian Optimization (Lee et al., 2020)
• Stochastic Optimization with Parametric Cost Function Approximations (III et al., 2017)
• A Rolling PID Control Approach and its Applications (zhou, 2016)
• Adjoint-based optimization for thrust performance of a three-dimensional pitching-rolling plate (Xu et al., 2018)
• Bound Tightening using Rolling-Horizon Decomposition for Neural Network Verification (Zhao et al., 10 Jan 2024)
• Advancing MG Energy Management: A Rolling Horizon Optimization Framework for Three-Phase Unbalanced Networks Integrating Convex Formulations (Cortés et al., 19 Mar 2025)
• Motion Planning for Object Manipulation by Edge-Rolling (Boroji et al., 11 Oct 2024)
• Enhancing Rolling Horizon Production Planning Through Stochastic Optimization Evaluated by Means of Simulation (Schlenkrich et al., 22 Feb 2024)
• A Multi-Agent-Based Rolling Optimization Method for Restoration Scheduling of Electrical Distribution Systems with Distributed Generation (Feng et al., 2018)
• Multi-finger Manipulation via Trajectory Optimization with Differentiable Rolling and Geometric Constraints (Yang et al., 23 Aug 2024)
• Uniform Rolling: An LSST Observing Cadence Offering Sufficient Survey Uniformity for Comprehensive Cosmological Analysis (Leistedt et al., 18 Dec 2025)
• A rolling-horizon dynamic programming approach for collaborative caching (Yang et al., 2019)
• A rolling horizon heuristic approach for a multi-stage stochastic waste collection problem (Spinelli et al., 23 May 2024)
• On the Optimal Control of a Rolling Ball Robot Actuated by Internal Point Masses (Putkaradze et al., 2017)
• Dynamic Rolling Horizon Optimization for Network-Constrained V2X Value Stacking of Electric Vehicles Under Uncertainties (Jiang et al., 13 Feb 2025)
• Rolling Optimization of Mobile Energy Storage Fleets for Resilient Service Restoration (Yao et al., 2019)
• Dynamic Rolling Horizon-Based Robust Energy Management for Microgrids Under Uncertainty (Hönen et al., 2023)
• Rolling in the Deep -- Hybrid Locomotion for Wheeled-Legged Robots using Online Trajectory Optimization (Bjelonic et al., 2019)
• Keep Rollin' - Whole-Body Motion Control and Planning for Wheeled Quadrupedal Robots (Bjelonic et al., 2018)
• Learning-Guided Rolling Horizon Optimization for Long-Horizon Flexible Job-Shop Scheduling (Li et al., 18 Feb 2025)