Anytime Feasibility in Computational Methods

• Anytime feasibility is a design property that guarantees a computational procedure returns a valid output at any interruption point.
• It spans multiple domains—such as constrained reinforcement learning, distributed optimization, MPC, and search—using mechanisms like state augmentation and forward-invariant safety.
• The approach emphasizes decoupling feasibility from performance, ensuring that partial solutions remain safe, incrementally improvable, and practically applicable.

Anytime feasibility is the property that a computational procedure remains valid at every interruption time, iteration, or resource budget, while further computation improves reward, optimality, robustness, or output quality. In the cited literature, the term is domain-dependent rather than uniform: in constrained reinforcement learning it denotes hard sample-path budget satisfaction at all times; in distributed optimization and energy management it denotes preservation of coupled balance constraints at every iterate; in model predictive control it denotes the ability to return a feasible, convergent control move regardless of optimization time; and in search, decoding, prediction, and Monte Carlo it denotes the availability of a complete valid output whenever computation stops (McMahan et al., 2023, Wu et al., 4 Aug 2025, Doostmohammadian, 2023, Bemporad et al., 2015, Hansen et al., 2011, Murray et al., 2016).

1. Core semantics across domains

Across the cited works, anytime feasibility appears in several recurring forms. One form is state-trajectory safety, where every realized trajectory must remain inside a hard feasible region. A second is iterate-wise constraint satisfaction, where each intermediate iterate of an algorithm must satisfy coupled resource or balance constraints. A third is interruption-time validity, where partial computation must already define a complete prediction, schedule, decoded sequence, or control action. A fourth is resource-budget correctness, where validity is defined relative to wall-clock, memory, or compute budgets rather than only to mathematical constraints (McMahan et al., 2023, Wu et al., 4 Aug 2025, Paige et al., 2014).

Setting	Feasibility notion	Representative work
Constrained RL	Budget never exceeded at any time, almost surely	(McMahan et al., 2023)
Distributed control/allocation	Coupled inequality/equality holds at every iterate	(Wu et al., 4 Aug 2025, Doostmohammadian, 2023, Doostmohammadian et al., 8 Sep 2025)
MPC	Feasible stabilizing plan always available	(Bemporad et al., 2015)
Search/prediction	Complete valid output available after interruption	(Hansen et al., 2011, Grubb et al., 2013, Lee et al., 2018, Xu, 2021)
Monte Carlo	Statistically valid output under real-time or memory budget	(Murray et al., 2016, Paige et al., 2014)
Temporal plans	Current robustness envelope already guarantees valid execution region	(Cashmore et al., 2019)
Scheduling	Schedule validity or acceptable response-time behavior maintained online	(Módos et al., 2018, 0801.4292, 0710.4737)

This diversity matters because feasibility is not synonymous with optimality. In the bi-objective Next Release Problem, the “anytime” property concerns maintaining a feasible, mutually non-dominated, well-spread set of solutions at any interruption time, not enforcing a physical safety constraint (Domínguez-Ríos et al., 2024). In crowd-sourced medical annotation with SAM, feasibility is split between the feasibility of sparse annotation by non-experts and the separate feasibility of using the resulting labels to train 3D models; the first is supported, whereas the second is not yet supported in the reported experiments (Kulkarni et al., 2024).

2. Formal characterizations of hard anytime feasibility

In anytime-constrained reinforcement learning, the underlying object is a finite-horizon constrained MDP $M=(S,A,P,R,H,s_0,C,B)$, and the hard feasibility condition is

$$\max_\pi \; \mathbb{E}^{\pi}_M\Big[\sum_{h=1}^H r_h(s_h,a_h)\Big] \quad\text{s.t.}\quad \mathbb{P}^{\pi}_M\Big(\forall t\in[H],\; \sum_{h=1}^t c_h \le B\Big)=1.$$

This is strictly stronger than expectation-constrained or chance-constrained formulations because the budget must never be exceeded at any time $t$ along any realized trajectory, with probability $1$ (McMahan et al., 2023).

In distributed resource allocation and energy management, anytime feasibility is formulated as invariance of a coupled global constraint. For inequality-coupled DRAPs, the requirement is $\sum_{i=1}^n A_i x_{i,k}\le \sum_{i=1}^n d_i$ for all iterations $k$; for the equality-coupled version it is $\sum_{i=1}^n A_i x_{i,k}= \sum_{i=1}^n d_i$ for all $k$ once feasible (Wu et al., 4 Aug 2025). In distributed energy resource management, the feasible manifold is

$a^\top z = b,$

where $z=[x_1;\dots;x_N;y_1;\dots;y_m]$ stacks production and reservation powers and $$\max_\pi \; \mathbb{E}^{\pi}_M\Big[\sum_{h=1}^H r_h(s_h,a_h)\Big] \quad\text{s.t.}\quad \mathbb{P}^{\pi}_M\Big(\forall t\in[H],\; \sum_{h=1}^t c_h \le B\Big)=1.$$0 encodes whether node $$\max_\pi \; \mathbb{E}^{\pi}_M\Big[\sum_{h=1}^H r_h(s_h,a_h)\Big] \quad\text{s.t.}\quad \mathbb{P}^{\pi}_M\Big(\forall t\in[H],\; \sum_{h=1}^t c_h \le B\Big)=1.$$1 is a generator or a storage unit; all-time feasibility means $$\max_\pi \; \mathbb{E}^{\pi}_M\Big[\sum_{h=1}^H r_h(s_h,a_h)\Big] \quad\text{s.t.}\quad \mathbb{P}^{\pi}_M\Big(\forall t\in[H],\; \sum_{h=1}^t c_h \le B\Big)=1.$$2 or $$\max_\pi \; \mathbb{E}^{\pi}_M\Big[\sum_{h=1}^H r_h(s_h,a_h)\Big] \quad\text{s.t.}\quad \mathbb{P}^{\pi}_M\Big(\forall t\in[H],\; \sum_{h=1}^t c_h \le B\Big)=1.$$3 at every continuous- or discrete-time step (Doostmohammadian, 2023). In distributed automatic generation control subject to ramp-rate limits, the same logic is applied to power mismatch balancing, together with per-step ramp-rate constraints such as $$\max_\pi \; \mathbb{E}^{\pi}_M\Big[\sum_{h=1}^H r_h(s_h,a_h)\Big] \quad\text{s.t.}\quad \mathbb{P}^{\pi}_M\Big(\forall t\in[H],\; \sum_{h=1}^t c_h \le B\Big)=1.$$4 (Doostmohammadian et al., 8 Sep 2025).

In anytime MPC, the feasibility notion is recursive and stabilizing. The finite-horizon problem includes standard dynamics and state-input constraints, a terminal-set condition $$\max_\pi \; \mathbb{E}^{\pi}_M\Big[\sum_{h=1}^H r_h(s_h,a_h)\Big] \quad\text{s.t.}\quad \mathbb{P}^{\pi}_M\Big(\forall t\in[H],\; \sum_{h=1}^t c_h \le B\Big)=1.$$5, and an explicit decrease condition

$$\max_\pi \; \mathbb{E}^{\pi}_M\Big[\sum_{h=1}^H r_h(s_h,a_h)\Big] \quad\text{s.t.}\quad \mathbb{P}^{\pi}_M\Big(\forall t\in[H],\; \sum_{h=1}^t c_h \le B\Big)=1.$$6

where $$\max_\pi \; \mathbb{E}^{\pi}_M\Big[\sum_{h=1}^H r_h(s_h,a_h)\Big] \quad\text{s.t.}\quad \mathbb{P}^{\pi}_M\Big(\forall t\in[H],\; \sum_{h=1}^t c_h \le B\Big)=1.$$7 is inherited from the previous feasible trajectory. This decouples performance optimization from convergence enforcement: any feasible solution of the constrained problem is sufficient to guarantee recursive feasibility and asymptotic convergence to the terminal set $$\max_\pi \; \mathbb{E}^{\pi}_M\Big[\sum_{h=1}^H r_h(s_h,a_h)\Big] \quad\text{s.t.}\quad \mathbb{P}^{\pi}_M\Big(\forall t\in[H],\; \sum_{h=1}^t c_h \le B\Big)=1.$$8 (Bemporad et al., 2015).

3. Mechanisms used to preserve feasibility during computation

A central mechanism is state augmentation by a sufficient statistic. In anytime-constrained cMDPs, Markov policies over the original state space are insufficient and can be arbitrarily suboptimal even for $$\max_\pi \; \mathbb{E}^{\pi}_M\Big[\sum_{h=1}^H r_h(s_h,a_h)\Big] \quad\text{s.t.}\quad \mathbb{P}^{\pi}_M\Big(\forall t\in[H],\; \sum_{h=1}^t c_h \le B\Big)=1.$$9 and $t$0. The sufficient statistic is cumulative cost, equivalently remaining budget. This yields an augmented MDP over states $t$1, with Bellman recursion restricted to actions that are safe almost surely. The resulting reduction is fixed-parameter tractable in the cost precision: with cost diversity $t$2, exact planning runs in $t$3, exact learning in $t$4, and if the cost precision is $t$5 with constant constraint dimension $t$6, planning and learning are polynomial (McMahan et al., 2023).

A second mechanism is forward-invariant safe-set enforcement. DanyRA formulates distributed resource allocation as a primal-dual method augmented by control barrier functions. Each agent projects its virtual iterate $t$7 onto a feasible affine set $t$8, thereby keeping the coupled constraint satisfied at every step. A virtual queue with minimum buffer $t$9 is then used to restore feasibility after external violations within finite time, trading asymptotic accuracy against violation robustness (Wu et al., 4 Aug 2025). Related distributed energy-management and AGC schemes enforce feasibility by antisymmetric Laplacian couplings: because the weighted edge contributions cancel pairwise under symmetric weights and odd sign-preserving nonlinearities, the aggregate balance variable remains invariant at all times (Doostmohammadian, 2023, Doostmohammadian et al., 8 Sep 2025).

A third mechanism is feasibility-first decoupling from performance optimization. In anytime MPC, a convex feasibility layer enforces the contractive terminal condition, while an outer optimization layer improves the cost when CPU time is available. If no time is available, the controller may shift the previously computed feasible plan and still retain convergence guarantees (Bemporad et al., 2015).

A fourth mechanism is incumbent-preserving refinement. In anytime end-effector trajectory tracking, the planner first returns an initial feasible trajectory, stores it as an incumbent, and thereafter only replaces it with lower-cost feasible trajectories. Guide paths and biased sampling accelerate time-to-first-feasible, while lazy edge validation and incremental graph search preserve feasibility during refinement (Wang et al., 5 Feb 2025). In temporal planning, Incremental Rectangular-Robustification enlarges a decoupled robustness envelope only after quantifier-free SMT checks certify that the new hyper-rectangle remains inside the exact robustness envelope; the current envelope is therefore executable at every intermediate time (Cashmore et al., 2019). In task-and-motion MDPs, partial policies are refined root-to-leaf, and already refined prefixes remain executable while the probability mass of covered outcomes grows over time (Srivastava et al., 2018).

4. Interruption-time validity in search, prediction, decoding, and Monte Carlo

In heuristic search, anytime feasibility is realized by maintaining an incumbent complete solution once the first goal is found. Anytime A* uses weighted evaluation $1$0 to find a first solution quickly, then continues search while preserving the incumbent cost $1$1 and a lower bound

$1$2

At any interruption time after the first solution, the algorithm can return the incumbent together with multiplicative and additive suboptimality bounds $1$3 and $1$4; under finite branching factor, positive minimal edge cost, nonnegative admissible heuristic, and existence of a solution, the algorithm terminates with the optimal solution (Hansen et al., 2011).

In structured prediction, feasibility means that the partial computation already defines a complete valid structured output. SpeedMachines maintains score vectors for all structured elements at every boosting step, and the softmax map enforces per-element simplex constraints $1$5. Thus, interruption never yields dangling or partially specified outputs; more weak updates improve the prediction while preserving feasibility (Grubb et al., 2013).

In neural prediction, anytime feasibility is defined relative to compute budget. For a sequence of sub-networks $1$6 with strictly increasing costs $1$7, the formal requirement is that for every budget $1$8, some slice $1$9 with $\sum_{i=1}^n A_i x_{i,k}\le \sum_{i=1}^n d_i$0 returns a valid prediction, with expected error non-increasing in $\sum_{i=1}^n A_i x_{i,k}\le \sum_{i=1}^n d_i$1 ideally. Vertical slicing of multi-branch networks into inclusive thin sub-networks satisfies this requirement empirically, and IS-ResNeXt enforces triangular sparsity so that all slice outputs are computable in a single forward pass, restoring interruptibility (Lee et al., 2018).

In decoding and Monte Carlo, the same pattern appears under computational budgets rather than abstract search depth. MCTS-based anytime decoding maintains $\sum_{i=1}^n A_i x_{i,k}\le \sum_{i=1}^n d_i$2 estimates and can greedily output a code-consistent sequence after any number of search rounds $\sum_{i=1}^n A_i x_{i,k}\le \sum_{i=1}^n d_i$3; more rounds improve BER and approximate maximum-likelihood sequence decoding more closely (Xu, 2021). In anytime Monte Carlo, a wall-clock budget induces state-dependent length bias in the terminal state distribution; a multiple-chain construction removes that bias by discarding the currently simulating chain and returning the waiting chains, which remain unbiased under the target distribution (Murray et al., 2016). The particle cascade makes sequential Monte Carlo asynchronous and anytime-feasible under a fixed memory budget by replacing barrier resampling with local asynchronous branching and multiplicity bookkeeping, allowing the algorithm to run indefinitely without exceeding a prescribed memory cap (Paige et al., 2014).

5. Scheduling, deadlines, and real-time notions of feasibility

In overloaded heterogeneous servers executing anytime algorithms, feasibility is not defined by hard deadline satisfaction alone. A schedule $\sum_{i=1}^n A_i x_{i,k}\le \sum_{i=1}^n d_i$4 is feasible if tasks are assigned to resources, start after arrival, and do not overlap on any resource. Response-time performance is then summarized by average normalized lateness

$\sum_{i=1}^n A_i x_{i,k}\le \sum_{i=1}^n d_i$5

When overload would make $\sum_{i=1}^n A_i x_{i,k}\le \sum_{i=1}^n d_i$6 at maximum quality $\sum_{i=1}^n A_i x_{i,k}\le \sum_{i=1}^n d_i$7, feasibility is effectively recovered by reducing requested qualities so that the recomputed lateness estimate $\sum_{i=1}^n A_i x_{i,k}\le \sum_{i=1}^n d_i$8 approaches zero. The paper introduces a global bisection heuristic and a per-task quality heuristic, both coupled to an EDD+MCT scheduler over heterogeneous resource speeds $\sum_{i=1}^n A_i x_{i,k}\le \sum_{i=1}^n d_i$9, and reports that both heuristics maintain lateness near zero even with estimation errors whose $k$0 quantile is around $k$1 (Módos et al., 2018).

This is a softer notion than schedulability in periodic real-time systems. For global periodic task scheduling on multiprocessors, exact feasibility is defined by all deadlines being met. Feasible schedules become periodic, with period equal to the hyperperiod $k$2, and exact tests are obtained by checking deadline satisfaction over a finite interval and verifying equality of scheduling states $k$3 and $k$4 at the repetition point (0801.4292). For uniprocessor EDF, exact feasibility is characterized by the demand bound function condition $k$5 over a finite feasibility interval, and the paper derives fast exact tests that refine sufficient superposition approximations only when needed, yielding an integrated hierarchy from sufficient screening to exact schedulability analysis (0710.4737).

A plausible implication is that “anytime feasibility” in real-time scheduling spans two distinct meanings: continuous maintenance of a valid schedule under ongoing overload adaptation, and progressive exact certification of schedulability as analysis time increases. The cited works instantiate both meanings, but they do not collapse to a single formal criterion (Módos et al., 2018, 0801.4292, 0710.4737).

6. Approximation, hardness, and common misconceptions

A recurring misconception is that anytime feasibility automatically admits efficient exact optimization. The strongest counterexample comes from anytime-constrained RL. Exact feasibility is NP-complete even when $k$6, $k$7, and both costs and rewards are deterministic non-negative integers; for $k$8, computing even a feasible solution is NP-hard, and for any $k$9 it is NP-hard to compute a feasible policy with additive or multiplicative $\sum_{i=1}^n A_i x_{i,k}= \sum_{i=1}^n d_i$0-optimality. The paper therefore shifts to approximation notions such as $\sum_{i=1}^n A_i x_{i,k}= \sum_{i=1}^n d_i$1-additive feasibility,

$\sum_{i=1}^n A_i x_{i,k}= \sum_{i=1}^n d_i$2

and $\sum_{i=1}^n A_i x_{i,k}= \sum_{i=1}^n d_i$3-relative feasibility, obtaining optimal-value policies with slack under bounded-support cost assumptions (McMahan et al., 2023).

A second misconception is that approximate envelopes or sparse outputs are interchangeable with exact feasibility. Decoupled robustness envelopes in temporal plans are conservative inner approximations: every parameter valuation inside the computed hyper-rectangle is safe, but valid valuations may also lie outside it (Cashmore et al., 2019). In the bi-objective Next Release Problem, anytime exactness refers to maintaining a feasible, non-dominated, decision-ready set throughout the search, not to having the full Pareto front at interruption; the point is precisely to preserve spread and feasibility before completion (Domínguez-Ríos et al., 2024).

A third misconception is that strong local or slice-wise agreement is sufficient for downstream feasibility. The medical-image annotation study reports that SAM- and MedSAM-generated annotations can achieve high slice Dice, yet nnU-Net models trained on those labels perform significantly worse than models trained on ground-truth labels, because 2D slice-wise masks assembled into 3D volumes lack spatial consistency. This directly separates “feasibility for sparse crowd-sourced annotation” from “feasibility for training-ready 3D supervision” (Kulkarni et al., 2024).

Finally, not every anytime method enforces hard constraints. The heterogeneous-resource scheduling work explicitly targets soft real-time behavior, with average normalized lateness near zero, and states that strict hard deadlines are not guaranteed (Módos et al., 2018). Anytime feasibility can therefore mean hard invariance, statistical correctness, complete-output validity, or decision readiness, depending on the problem class.

7. Cross-domain design principles

The cited literature suggests several recurring design principles. One is encode feasibility as an invariant object rather than a terminal by-product. In RL this object is the augmented cost state; in DanyRA and distributed energy management it is the coupled feasible manifold; in MPC it is the contractive constraint plus terminal set; in temporal planning it is the currently certified robustness envelope (McMahan et al., 2023, Wu et al., 4 Aug 2025, Doostmohammadian, 2023, Bemporad et al., 2015, Cashmore et al., 2019).

A second principle is maintain an incumbent feasible artifact and improve it monotonically. Anytime A* maintains an incumbent solution and tighter lower bounds; graph-based trajectory tracking maintains an incumbent feasible path; task-and-motion MDPs maintain executable prefixes; NRP anytime algorithms maintain a growing feasible non-dominated archive (Hansen et al., 2011, Wang et al., 5 Feb 2025, Srivastava et al., 2018, Domínguez-Ríos et al., 2024).

A third principle is separate feasibility from performance whenever optimization is expensive or hard. The MPC construction makes convergence independent of how much performance optimization is completed in a sampling interval. DanyRA enforces coupled feasibility independently of exact convergence speed. Anytime-constrained RL separates exact feasibility from approximate optimality when hardness rules out simultaneous guarantees (Bemporad et al., 2015, Wu et al., 4 Aug 2025, McMahan et al., 2023).

A fourth principle is use resource-aware surrogates only when they preserve the relevant validity notion. Weighted heuristics preserve complete-solution validity in search, but only after a first goal is found. Local branching in particle cascades preserves unbiasedness and memory bounds. Vertical slicing preserves valid predictions for any compute budget. Where such surrogates do not preserve the required downstream object, as in 2D-to-3D annotation transfer, reported quality can be misleading (Hansen et al., 2011, Paige et al., 2014, Lee et al., 2018, Kulkarni et al., 2024).

Taken together, these works portray anytime feasibility not as a single theorem but as a family of design doctrines for computation under interruption, uncertainty, and resource limitation. The unifying requirement is that partial computation must already satisfy the problem’s operative notion of validity; the technical content lies in how that validity is encoded, preserved, certified, and refined.