Rolling-Horizon Optimization (RHO)

• Rolling-Horizon Optimization (RHO) is a sequential decision framework that solves a finite-horizon problem at each step and re-optimizes as new information becomes available.
• It integrates deterministic, stochastic, and robust models to handle forecast uncertainty and ensure smooth inter-window coupling across various applications.
• RHO improves computational efficiency by decomposing long-horizon problems into manageable subproblems while mitigating the risks of myopic decisions through adaptive forecasting.

Rolling-Horizon Optimization (RHO) is a sequential decision procedure in which, at each decision epoch, an optimization problem is solved over a finite future window, only the first-stage action is implemented, new information is observed, and the horizon is shifted forward. In control settings this receding-horizon procedure is equivalent to model predictive control when future states are predicted explicitly from a system model, while in stochastic programming and large-scale mathematical optimization it functions as a tractability device for replacing a single long-horizon problem by a sequence of smaller but coupled subproblems (Papaioannou et al., 8 Apr 2025, Siddig et al., 2021).

1. Canonical formulation and policy interpretation

A general stochastic control formulation used in the RHO literature specifies a state $S_t$, decision $x_t\in\mathcal X_t$, exogenous information $W_{t+1}$, transition map $S_{t+1}=S^M(S_t,x_t,W_{t+1})$, and per-period cost $C(S_t,x_t)$. The corresponding base model is

$$\min_{\pi\in\Pi}\;\mathbb E^\pi\Bigl[\sum_{t=0}^T C\bigl(S_t,X_t^\pi(S_t)\bigr)\;\Big|\;S_0\Bigr],$$

subject to

$$S_{t+1}=S^M\bigl(S_t,X_t^\pi(S_t),W_{t+1}\bigr),\quad X_t^\pi(S_t)\in\mathcal X_t.$$

Within this viewpoint, RHO is not the base problem itself but a policy class for solving it: at time $t$, one solves a truncated lookahead problem on a smaller horizon, applies the first action, and re-optimizes after the system evolves (III et al., 2017).

For finite-horizon multistage stochastic programming, the truncated lookahead problem at stage $t$ with horizon length $\tau$ is written as $x_t\in\mathcal X_t$0, where the terminal cost-to-go $x_t\in\mathcal X_t$1 is set to zero. The resulting RH policy implements only the first-stage decision $x_t\in\mathcal X_t$2, then rolls forward one period and repeats (Siddig et al., 2021). In scenario-based variants, this first stage is typically coupled by non-anticipativity constraints. In stochastic home energy management, for example, first-hour battery decisions satisfy $x_t\in\mathcal X_t$3, $x_t\in\mathcal X_t$4, and $x_t\in\mathcal X_t$5 for all scenarios $x_t\in\mathcal X_t$6, so that the action executed at the current time is scenario-independent (Lemos-Vinasco et al., 2022).

This formulation makes RHO simultaneously a decision rule, a decomposition strategy, and a way of embedding updated measurements or forecasts into repeated optimization.

2. Deterministic, stochastic, robust, and multi-horizon variants

The inner model solved inside an RHO loop varies substantially across the literature. A classical deterministic rolling horizon, often written with point forecasts $x_t\in\mathcal X_t$7, solves a short-window deterministic program and implements only the current action. In practice, however, practitioners modify such models with buffer stocks, schedule slack, reserve requirements, or forecast corrections. A central reinterpretation is that these “deterministic” remedies are parametric cost function approximations (CFAs), for example

$x_t\in\mathcal X_t$8

where $x_t\in\mathcal X_t$9 adjusts the objective and right-hand sides and is tuned in the stochastic base model, typically by simulation (III et al., 2017).

A second family uses explicit stochastic lookahead models. In residential energy management, the rolling window is a two-stage stochastic MILP that minimizes expected electricity cost over a 24-hour horizon with 100 demand scenarios drawn from online multivariate probabilistic-load-forecast methods, each with probability $W_{t+1}$0 (Lemos-Vinasco et al., 2022). In production planning, the rolling subproblem is a two-stage capacitated lot-sizing model with here-and-now lot sizes for $W_{t+1}$1 and recourse lot sizes, inventories, and backlogs for $W_{t+1}$2, embedded in discrete-event simulation (Schlenkrich et al., 2024).

A third family is robust. In microgrid scheduling, uncertain loads, photovoltaic generation, EV demand, and market prices are represented through box and budget uncertainty sets, and a static robust counterpart is repeatedly solved over a moving window (Hönen et al., 2023). Related energy-management work combines an online scheduling layer with robust optimization, where the value of launching a new RHO iteration depends on forecast-driven information gain rather than a fixed timetable (Hönen et al., 2023).

A fourth family is deterministic re-optimization with learned point forecasts rather than explicit scenarios. In network-constrained V2X coordination, the rolling subproblem is an MIQP solved hourly on the shrinking horizon $W_{t+1}$3, using realized data up to $W_{t+1}$4 and point forecasts from the GRU-EN-TFD model for later periods (Jiang et al., 13 Feb 2025).

A fifth family is multistage multi-horizon stochastic programming. In domestic renewable energy system design, strategic investment stages and short-term operational subtrees are linked in a single tactical multi-horizon tree, and a rolling-horizon matheuristic solves submodels containing both a master investment problem and operational recourse, optionally with first- or second-order stochastic dominance constraints that preserve time consistency (Micheli et al., 21 May 2025).

Taken together, these variants show that RHO is compatible with deterministic LPs, MILPs, MIQPs, robust counterparts, and scenario-based stochastic programs; the rolling mechanism is separate from the precise uncertainty model used within each window.

3. Horizon length, overlap, and inter-window coupling

The central design question in RHO is how far to look ahead and how much of the computed plan to execute before re-optimization. In infinite-horizon discounted multistage stochastic programming with bounded stage costs $W_{t+1}$5 and discount factor $W_{t+1}$6, the lookahead policy $W_{t+1}$7 satisfies

$W_{t+1}$8

and an $W_{t+1}$9-sufficient forecast horizon is therefore

$S_{t+1}=S^M(S_t,x_t,W_{t+1})$0

For finite horizons, Siddig, Song and Khademi propose an ADP-style heuristic that learns a state-dependent horizon map $S_{t+1}=S^M(S_t,x_t,W_{t+1})$1, so that the lookahead depth varies by roll rather than remaining fixed (Siddig et al., 2021).

Overlap is the standard mechanism for preserving continuity across windows. In offline pickup-and-delivery with time windows, the day is decomposed into overlapping windows $S_{t+1}=S^M(S_t,x_t,W_{t+1})$2; each subproblem is solved on the active requests in the window, but only the first $S_{t+1}=S^M(S_t,x_t,W_{t+1})$3 minutes are frozen before the horizon advances. This overlapping structure provides the boundary stitching that makes temporal decomposition viable for narrow-window PDPTW instances (Kim et al., 2023). In flexible job-shop scheduling, the analogous parameters are planning window $S_{t+1}=S^M(S_t,x_t,W_{t+1})$4, execution step $S_{t+1}=S^M(S_t,x_t,W_{t+1})$5, and overlap size $S_{t+1}=S^M(S_t,x_t,W_{t+1})$6; at each iteration the next $S_{t+1}=S^M(S_t,x_t,W_{t+1})$7 operations are scheduled, but only the first $S_{t+1}=S^M(S_t,x_t,W_{t+1})$8 are executed (Li et al., 18 Feb 2025).

Other domains formalize inter-window coupling through state carryover or fixed first-stage commitments. In rolling lot-sizing, here-and-now quantities $S_{t+1}=S^M(S_t,x_t,W_{t+1})$9 are fixed across scenarios for $C(S_t,x_t)$0, while later decisions remain adjustable (Schlenkrich et al., 2024). In multi-horizon stochastic MILP for domestic energy system design, once a window has been solved, first-stage investments are fixed and battery state is carried into the next iteration (Micheli et al., 21 May 2025). In healthcare operating-theatre scheduling, week 1 is implemented and frozen, then weeks 2 through $C(S_t,x_t)$1 are replanned under rolling-horizon “no-move-right” constraints (Spratt et al., 2018). In stochastic waste collection, inventory levels $C(S_t,x_t)$2 are passed from one subproblem to the next as a warm start (Spinelli et al., 2024).

These mechanisms control the basic RHO trade-off: larger horizons reduce myopia but increase subproblem size, whereas smaller horizons improve responsiveness and runtimes but amplify truncation effects.

4. Forecasting, learning, and information-driven replanning

A defining feature of contemporary RHO is the tight coupling between forecasting and optimization. In stochastic home energy management, two online probabilistic-load-forecast methods are used—RLS-Free and Copula-Free—and each hourly update draws 100 net-demand scenarios for the 24-hour horizon, with no scenario reduction (Lemos-Vinasco et al., 2022). In production planning, forecast evolution is modeled through additive MMFE dynamics, and customer behaviors are parameterized as Type A, Type B, and Type C, corresponding respectively to stable, single late-update, and periodic-update forecasts (Schlenkrich et al., 2024). In collaborative caching, demand prediction uses an auto-regressive model $C(S_t,x_t)$3, and the rolling policy optimizes against cumulative predicted demand over an $C(S_t,x_t)$4-step lookahead (Yang et al., 2019).

Recent work increasingly learns either the forecasts themselves or the structure of the rolling procedure. In V2X value stacking, the GRU-EN-TFD forecaster predicts building load, PV output, and EV arrivals; the RHO then re-solves an MIQP each hour on the remaining part of the day (Jiang et al., 13 Feb 2025). In dynamic microgrid management, a knapsack-type model selects the best iteration start times by quantifying the expected value of refined PV forecasts and realized EV arrivals, so that re-optimization is concentrated where information changes most (Hönen et al., 2023). In threshold-based online rolling horizon control, an offline longest-path problem produces contribution values $C(S_t,x_t)$5, these are collapsed into a threshold $C(S_t,x_t)$6, and a new iteration is launched online whenever the estimated contribution exceeds a dynamic threshold $C(S_t,x_t)$7; with a median threshold, the resulting online rule is shown to be $C(S_t,x_t)$8-competitive (Hönen et al., 2023).

Learning also appears inside the optimization loop itself. Zhao et al. use a rolling-horizon MILP decomposition for neural network verification, where overlapping layer subgraphs are processed sequentially to tighten pre-activation bounds (Zhao et al., 2024). Li et al. propose L-RHO for long-horizon flexible job-shop scheduling: a neural network predicts which overlapping operations are unlikely to change machine assignment between consecutive windows, and those assignments are fixed in the next subproblem, reducing redundant re-optimization (Li et al., 18 Feb 2025).

This body of work suggests a shift from fixed-calendar rolling schemes toward information-adaptive rolling schemes in which forecasting, scenario construction, and variable fixing are themselves optimized.

5. Representative applications and reported performance

Empirical results are strongly domain-specific, but the literature reports recurrent gains when uncertainty is material, horizons are long, or monolithic formulations are intractable.

Domain	Rolling subproblem	Reported findings
Energy storage management	Parametric CFA within deterministic LP lookahead	Under perfect forecasts, the unparameterized RHO is optimal; as forecast noise increases, CFA methods outperform the benchmark by 13–26%, and “lookup-table” and “exponential” forms give up to ~26% improvement for large $C(S_t,x_t)$9 (III et al., 2017)
Stochastic home energy management	Two-stage stochastic MILP, $$\min_{\pi\in\Pi}\;\mathbb E^\pi\Bigl[\sum_{t=0}^T C\bigl(S_t,X_t^\pi(S_t)\bigr)\;\Big|\;S_0\Bigr],$$0, 100 scenarios	RLS-SP yields up to 10–20% lower costs than naive in winter and autumn; “Naive+RLS-SP” gives an extra $$\min_{\pi\in\Pi}\;\mathbb E^\pi\Bigl[\sum_{t=0}^T C\bigl(S_t,X_t^\pi(S_t)\bigr)\;\Big|\;S_0\Bigr],$$1 cost saving; RLS-SP is within 3–6% of the perfect-information bound, but naive outperforms pure stochastic RHO in spring and summer (Lemos-Vinasco et al., 2022)
Production planning	Scenario-based stochastic lot sizing in simulation	For highly nervous Type C forecasts, stochastic optimization cuts 40–60% vs. MRP and 5–25% vs. deterministic; at $$\min_{\pi\in\Pi}\;\mathbb E^\pi\Bigl[\sum_{t=0}^T C\bigl(S_t,X_t^\pi(S_t)\bigr)\;\Big|\;S_0\Bigr],$$2 and 98% utilization, stochastic beats deterministic by ~14%; best number of scenarios is ≈30, with solve times ~15–20 s (Schlenkrich et al., 2024)
Robust microgrid scheduling	Dynamic rolling robust LP	The dynamic rolling horizon model outperforms the classical version by up to 57% in costs and increases the local use of PV by up to 11% (Hönen et al., 2023)
Distributed UAV coverage	Rolling-horizon distributed MIP	With $$\min_{\pi\in\Pi}\;\mathbb E^\pi\Bigl[\sum_{t=0}^T C\bigl(S_t,X_t^\pi(S_t)\bigr)\;\Big|\;S_0\Bigr],$$3, mission completion is roughly 15% faster than myopic coverage and still 6× more computationally efficient than centralized RHO; per-step solve times are ∼0.2–1 s (Papaioannou et al., 8 Apr 2025)
Neural network verification	Rolling-horizon OBBT MILPs on layer subgraphs	OBBT-RH verifies 80 of 90 instances and is 1.25× faster overall than full OBBT+MIP while preserving near-OBBT bound quality (Zhao et al., 2024)
Online energy management	Threshold-scheduled robust RHO	Total electricity costs are reduced by more than 85% compared to a classical rolling horizon framework and by more than 50% compared to a dynamic offline rolling horizon framework (Hönen et al., 2023)
Long-horizon flexible job-shop scheduling	Learning-guided restricted CP-SAT subproblems	L-RHO accelerates RHO by up to 54% and improves objective value by up to 21% relative to Default RHO (Li et al., 18 Feb 2025)

Across these studies, the strongest gains typically occur when forecast revisions are frequent, constraints are tight, or the baseline is either myopic or static. Reported exceptions are equally informative: perfect forecasts can eliminate the need for corrective parameterization, and finite-window myopia can make simple fallback policies preferable in low-volatility regimes.

6. Limitations, misconceptions, and current research directions

A persistent misconception is that the main methodological divide is between “deterministic rolling horizon” and “stochastic programming.” The parametric-CFA perspective argues instead that both are policies for a stochastic base model, differing mainly in where approximation is placed: scenario trees and recourse structure on one side, tunable objective and constraint modifications on the other (III et al., 2017).

Myopia remains the principal technical limitation. In waste collection, Theorem 3.2 shows that when $$\min_{\pi\in\Pi}\;\mathbb E^\pi\Bigl[\sum_{t=0}^T C\bigl(S_t,X_t^\pi(S_t)\bigr)\;\Big|\;S_0\Bigr],$$4 there exist feasible instances for which $$\min_{\pi\in\Pi}\;\mathbb E^\pi\Bigl[\sum_{t=0}^T C\bigl(S_t,X_t^\pi(S_t)\bigr)\;\Big|\;S_0\Bigr],$$5 although the full multistage model is feasible; empirically, $$\min_{\pi\in\Pi}\;\mathbb E^\pi\Bigl[\sum_{t=0}^T C\bigl(S_t,X_t^\pi(S_t)\bigr)\;\Big|\;S_0\Bigr],$$6 produced infinite or very large profit loss in several small instances, whereas $$\min_{\pi\in\Pi}\;\mathbb E^\pi\Bigl[\sum_{t=0}^T C\bigl(S_t,X_t^\pi(S_t)\bigr)\;\Big|\;S_0\Bigr],$$7 traded ∼29.8% average profit gap for 90–98% CPU-time savings (Spinelli et al., 2024). Similar finite-window effects appear in residential energy management, where pure stochastic RHO can misjudge the long-term value of PV self-consumption in spring and summer (Lemos-Vinasco et al., 2022).

Safeguards against such pathologies are an active research direction. In situationally aware multi-tier load restoration, safeguarded constraints are added so that each segment remains above a $$\min_{\pi\in\Pi}\;\mathbb E^\pi\Bigl[\sum_{t=0}^T C\bigl(S_t,X_t^\pi(S_t)\bigr)\;\Big|\;S_0\Bigr],$$8-fraction of its optimal restoration potential, explicitly mitigating the myopia of plain rolling horizon control (Shi et al., 2 Oct 2025). In large multi-horizon stochastic MILPs, lower-bound schemes such as Strategic Wait-and-See, Scenario Multistage Grouping, and Scenario Multistage Clustering are used to assess the optimality gap of rolling-horizon heuristics; with $$\min_{\pi\in\Pi}\;\mathbb E^\pi\Bigl[\sum_{t=0}^T C\bigl(S_t,X_t^\pi(S_t)\bigr)\;\Big|\;S_0\Bigr],$$9, $$S_{t+1}=S^M\bigl(S_t,X_t^\pi(S_t),W_{t+1}\bigr),\quad X_t^\pi(S_t)\in\mathcal X_t.$$0, and $$S_{t+1}=S^M\bigl(S_t,X_t^\pi(S_t),W_{t+1}\bigr),\quad X_t^\pi(S_t)\in\mathcal X_t.$$1, the SFR3 matheuristic achieved solutions within 0.05–0.2% of optimum on small and medium instances and within 0.7% of the best lower bound on large cases (Micheli et al., 21 May 2025).

A second limitation is computational scaling. Longer horizons tighten bounds and improve non-myopic behavior, but they enlarge the MILP or MIQP rapidly. In rolling-horizon verification, larger layer horizons yield diminishing returns while subproblem costs grow super-linearly (Zhao et al., 2024). In distributed coverage control, longer horizons add $$S_{t+1}=S^M\bigl(S_t,X_t^\pi(S_t),W_{t+1}\bigr),\quad X_t^\pi(S_t)\in\mathcal X_t.$$2 binary variables (Papaioannou et al., 8 Apr 2025). This has motivated decomposition, parallelism, online scheduling of iteration times, and learning-guided fixing of stable decisions.

Current work therefore concentrates on three complementary directions: stronger theoretical guidance for selecting horizon length, more adaptive policies for deciding when to re-optimize, and hybrid schemes that combine rolling windows with simulation, learning, or lower-bound certification. The cumulative evidence does not treat RHO as a single algorithmic template; rather, it identifies a broad family of sequential optimization policies whose effectiveness depends on how horizon truncation, uncertainty representation, and inter-window coupling are engineered for the underlying problem class.