Deterministic Restoration Plans

• Deterministic restoration plans are precisely defined action sequences that restore system operations after disruptions with no internal randomness.
• They employ rigorous mathematical models and deterministic algorithms (e.g., MILP, rule-based heuristics) to guarantee predictable, verifiable outcomes.
• Applications span power grid recovery and image restoration, offering robust performance, reproducibility, and operational compliance.

A deterministic restoration plan is a precisely specified, reproducible sequence of actions designed to recover system functionality after a disruptive event such as a blackout, natural disaster, or severe degradation. Deterministic plans are characterized by their lack of internal randomness: for any given input scenario—including system topology, resource state, available budget, and parameter settings—they produce a unique, unambiguous restoration sequence or action schedule. This approach is fundamental both in infrastructure resilience planning (e.g., power grid, distribution systems) and in computational restoration tasks (e.g., image recovery, inverse problems in scientific computing). Deterministic restoration plans are widely adopted because they deliver predictability, accountability, and guarantee reproducibility of the restoration process across repeated executions or simulations.

1. Mathematical Formulation and Problem Structure

Deterministic restoration planning problems are typically formulated as discrete-time, resource-constrained scheduling or mixed-integer programming (MIP) models. The mathematical structure varies by domain but shares several features:

• Variables:
  • Binary indicators for action status (e.g., $x_{l}^t$ = 1 if component $l$ is restored at time $t$).
  • State variables tracking system status, energization (e.g., $u_{n}^t$ = 1 if node $n$ energized at $t$).
  • Auxiliary variables for cumulative cost, resilience, or reward.
• Objective functions:
  • Power networks: Minimize total harm or unserved load, maximize restored load, minimize makespan or overall restoration time (1711.02205, Chopra et al., 2022, Jang et al., 2020).
  • Image restoration: Maximize perceptual or fidelity metrics (e.g., PSNR, SSIM, DeQA-Score) by choosing optimal tool sequences (Lu et al., 21 Dec 2025, Liu et al., 18 Nov 2025).
  • Generalized resource scheduling: Maximize total reward collected under budget (e.g., cost-constrained mTSP formulations) (Wei et al., 2024).
• Constraints:
  • Precedence and connectivity (e.g., network radiality, path-continuity).
  • Resource limitations (repair crew limits, switching actions/step, tool usage budget).
  • Physical or operational constraints (power flow balance, voltage/frequency/thermal limits, data-fidelity in inverse problems).

Key examples:

• The inner problem from (1711.02205) is an MILP with explicit sequencing, connectivity, time-tracking, and harm-linearization constraints.
• Multi-stage models represent restoration over a time horizon ($T$), linking state across stages to enforce monotonic resource addition and system progression (Rhodes et al., 2020, Jang et al., 2020, Chopra et al., 2022).

2. Deterministic Restoration Algorithms and Execution

Solution techniques for deterministic restoration plans include exact optimization, greedy heuristics, and rule-based automated procedures.

Exact and MILP-based approaches:

• Solve the tightly formulated MILP or MISOCP, resulting in a unique plan for every fixed parameterization (1711.02205, Wei et al., 2024, Poudel et al., 2020, Virginillo et al., 6 Nov 2025).
• Several frameworks, e.g., PowerModelsRestoration.jl, natively support deterministic restoration ordering with connectivity and resource constraints across periods (Rhodes et al., 2020).

Greedy and combinatorial heuristics:

• Multi-run greedy approaches efficiently approximate the MILP optimum with deterministic tie-breaking and efficiency-based prioritization (e.g., outtree-merging for repair sequencing) (1711.02205).
• Deterministic multi-agent pathfinding or resource allocation encapsulated as deterministic variants of multi-TSP and knapsack problems (Wei et al., 2024).

Rule-based and prioritization algorithms:

• Selection and action order fully determined by deterministic priority rules (generator picking order, load incremental order, resource buffer thresholds) (Jang et al., 2020, Mate et al., 2021).
• Pre-specified user criteria map arbitrary input data to unique action sequences, ensuring the absence of randomization (Jang et al., 2020).

Pseudocode and workflow reproducibility:

• For fixed inputs, the workflow from preprocessing through action sequencing is completely specified, with no probabilistic branching. Representative pseudocode is given in (Jang et al., 2020, Lu et al., 21 Dec 2025), and (Poudel et al., 2020).

3. Key Principles: Robustness, Hardening, and Tradeoffs

Deterministic restoration planning extends beyond basic sequencing to include robustness and system hardening strategies:

• Two-stage robust approaches: Separate the plan into (1) inner deterministic sequencing given a damage scenario, and (2) outer robustification or hardening via deterministic surrogate optimization (Jensen’s inequality, convex replacements for repair time uncertainty) (1711.02205).
• Resilience metrics: Quantify operability via resilience integrals, operability trajectories, or cumulative harm, and design plans to optimize these deterministic metrics (1711.02205).
• Perception–robustness tradeoff in deterministic methods: In image restoration and inverse problems, high-fidelity deterministic plans (low joint perceptual error) necessarily increase adversarial susceptibility, with the Lipschitz constant growing as the plan approaches perfect consistency and realism (Ohayon et al., 2023). Any practical plan must select a point along this tradeoff boundary, balancing perceptual gains with required stability.
• Posterior exploration via adversarial direction: The inherent instability of high-fidelity deterministic restorers can be systematically exploited to generate diverse outputs by input perturbation, mimicking stochastic posterior sampling (Ohayon et al., 2023).

4. Applications Across Domains

Deterministic restoration plans are foundational in several domains:

Power Grids and Infrastructure

• Distribution system restoration: MILP-based sequencing of repairs, deterministic robust hardening, and combinatorial crew-scheduling (1711.02205, Wei et al., 2024, Chopra et al., 2022, Fu et al., 2021).
• Transmission and mixed networks: PowerModelsRestoration.jl enables deterministic multi-period planning with exact or convex relaxations (Rhodes et al., 2020). Large-scale systems employ deterministic parallelization or resource-constrained sectionalization (Chopra et al., 2022).
• Service restoration with operational constraints: Two-stage MILPs ensure all intermediate steps maintain feasibility—radiality, voltage, frequency, and CLPU dynamics—across the entire deterministic switching sequence (Poudel et al., 2020, Virginillo et al., 6 Nov 2025).

Machine Perception and Image Restoration

• Label-free agentic restoration: RL agents produce deterministic tool-calling sequences via policy optimization, with inference performed greedily and uniquely for each input (Lu et al., 21 Dec 2025). Plans are invariant under repeated runs for the same degraded input and model.
• Frequency- and semantic-aware plans: Restoration is controlled by deterministic plans generated via frozen MLLMs under fixed decoding strategies, routing expert adapters deterministically in the downstream restoration network (Liu et al., 18 Nov 2025).
• Deterministic diffusion and inverse methods: Purely deterministic plug-and-play solvers employ gradient flows and fixed denoising networks, yielding a unique trajectory for restoration with given parameters (Wang et al., 3 Mar 2025, Chen et al., 2023).

5. Determinism, Reproducibility, and Scalability

A distinguishing feature of deterministic restoration plans is strict reproducibility for fixed inputs, configuration, and seed settings:

Approach/Domain	Determinism Mechanism	Reproducibility
MILP/ILP/convex optimization	Fixed input + solver parameterization	Unique solution unless solver randomizes tie-break
Rule-based heuristic	Fully specified rulechain, no RNG	One plan per input data and user parameters
Greedy multi-run algorithms	Deterministic sorting and selection	Plan invariant to repeated runs
RL/planner-based AI restoration	Greedy policy (argmax), tau=0 decoding, no sampling	One deterministic trajectory per input-image

Solvers such as Gurobi or CPLEX are made deterministic via fixed random seeds, and heuristic or rule-based methods are devoid of stochastic branching. This is essential for grid-operational regulatory compliance and for benchmarking algorithm research (1711.02205, Wei et al., 2024).

Large-scale restoration planning employs tailored deterministic heuristics, tree and graph reduction, or randomized outer loops that yield deterministic outputs once a sample is fixed (Chopra et al., 2022).

6. Selected Computational Results and Performance Highlights

Key empirical findings from deterministic restoration studies include:

• Distribution system resilience: On benchmark grids (IEEE 13, 37, 8500 nodes), greedy deterministic heuristics recover >95% optimality in seconds to minutes where MILP is intractable (1711.02205).
• Power grid large networks: Deterministic ILP and randomized outer wrappers produce high-quality restoration plans on systems up to 2000 buses, with plan quality gaps <5% and execution times of order minutes (Chopra et al., 2022).
• Image restoration: Deterministic RL-based plans (SimpleCall) maintain <0.7s inference latency regardless of input complexity and match or outperform SOTA full- and no-reference quality measures (Lu et al., 21 Dec 2025). Frequency-aware deterministic planners avoid sampling-induced variance and yield reproducible, semantically interpretable execution pipelines (Liu et al., 18 Nov 2025).
• Inverse problems: Non-asymptotic convergence guarantees are derived for deterministic DDIM-type samplers, with polynomial scaling in error tolerance and system dimension (Chen et al., 2023).

7. Practical Guidelines and Implementation Considerations

For successful deployment of deterministic restoration planning:

• Input validation: Exact knowledge of system status, available resources, and parameter configuration are preconditions for deterministic plan generation.
• Plan verification: Test plans for all operating limits and resilience constraints across simulated or historical scenarios to guarantee safety and feasibility at every stage (Poudel et al., 2020).
• Solver configuration: Use fixed seeds and logging options to ensure deterministic behavior in commercial solvers (Wei et al., 2024).
• Plan updates: For real-time adaptation, deterministic re-planning is triggered on significant scenario changes; receding-horizon approaches ensure responsiveness without loss of determinism at each planning instant (Fu et al., 2021).

In summary, deterministic restoration plans constitute the methodological backbone for resilient, reproducible, and auditable recovery operations across energy systems, smart infrastructure, and computational restoration tasks, with rigorous mathematical structure and diverse algorithmic realizations across application domains (1711.02205, Chopra et al., 2022, Lu et al., 21 Dec 2025, Jang et al., 2020).