DanyRA: Distributed Anytime-Feasible Allocation

• DanyRA is a distributed resource allocation approach that exactly preserves the global coupling constraint at every iteration.
• It employs nonlinear, consensus-based primal dynamics along with robust control mechanisms to handle delays, failures, and saturation.
• Its invariant-preserving design is crucial for reliable applications like energy dispatch, CPU scheduling, and networked resource management.

Distributed Anytime-Feasible Resource Allocation (DanyRA) denotes a class of distributed resource allocation methods whose defining invariant is exact satisfaction of the global coupling constraint throughout the iterative process, rather than only at convergence. In the canonical form, agents minimize a separable convex objective such as

$$\min_{x \in \mathbb{R}^n} \quad F(x)=\sum_{i=1}^n f_i(x_i) \quad \text{s.t.} \quad \sum_{i=1}^n x_i=b,$$

over a networked communication graph, with the additional requirement that the iterate sequence remain feasible at every step. Later work explicitly adopts the DanyRA terminology, while earlier papers used the closely related phrase “all-time feasible” for the same structural property (Doostmohammadian et al., 21 Oct 2025, Doostmohammadian et al., 2024).

1. Problem class and formal scope

The core DanyRA problem is distributed resource allocation over a set of agents $V=\{1,\dots,n\}$ communicating on an undirected, possibly time-varying graph $\mathcal{G}(k)=(V,E(k))$ with symmetric weights $W(k)$. Each agent $i$ controls a scalar or vector decision variable and a local objective $f_i$, while the network as a whole must satisfy a coupled equality or inequality constraint. In the most common scalar form, the coupled constraint is $\sum_i x_i=b$; in more general formulations it becomes $\sum_i A_i x_i \le b$ or $\sum_i A_i x_i=b$, with full row-rank $A_i$ and linear coupling maps $V=\{1,\dots,n\}$0 (Doostmohammadian et al., 21 Oct 2025, Wu et al., 4 Aug 2025).

Strict convexity and smoothness are standard assumptions. One recurrent condition is $V=\{1,\dots,n\}$1, which supports Lyapunov descent and step-size bounds. Local box or capacity constraints are typically not imposed by hard projections in the baseline primal schemes; instead they are incorporated via exact or approximate penalty/barrier constructions, including piecewise-polynomial penalties and smooth barrier-like penalties such as $V=\{1,\dots,n\}$2 (Doostmohammadian et al., 21 Oct 2025). Earlier formulations also admit convex local sets through barrier functions on the interior of the feasible domain, transforming local constraints into smooth objective terms while preserving the coupled feasibility invariant (Wu et al., 2021).

This problem class is broader than a single algorithmic template. It includes first-order consensus-based primal dynamics for equality-coupled convex programs, momentum-augmented and signum-accelerated variants, delay-tolerant nonlinear gradient-Laplacian methods, and more recent primal-dual safe-control constructions for inequality-coupled problems (Doostmohammadian et al., 2021, Doostmohammadian et al., 8 Mar 2025, Wu et al., 4 Aug 2025). A parallel line of work in software-defined networking also realizes the same anytime-feasible principle by producing a feasible bandwidth allocation at every ADMM iteration (Allybokus et al., 2017, Allybokus et al., 2018).

2. The anytime-feasible invariant

The defining DanyRA property is the invariant

$V=\{1,\dots,n\}$3

or its generalized coupled-constraint analogue. In the primal Laplacian-based family, this invariance is enforced structurally rather than by projection. A representative delay-free update is

$V=\{1,\dots,n\}$4

where $V=\{1,\dots,n\}$5 and $V=\{1,\dots,n\}$6 are odd, sign-preserving, and sector-bounded. Because the weights are symmetric and the edge terms are antisymmetric, the network sum of increments is exactly zero, so feasibility is preserved from any feasible initialization (Doostmohammadian et al., 21 Oct 2025).

This algebraic cancellation mechanism recurs across the literature. “Accelerated Distributed Allocation” proves that the continuous- and discrete-time signum-based primal dynamics are all-time feasible because the weighted pairwise nonlinear gradient exchanges cancel in the aggregate (Doostmohammadian et al., 2024). The same principle underlies earlier nonlinear-agent and energy-management formulations, where odd sign-preserving mappings model saturation, quantization, or sign-based feedback without destroying conservation of the coupling constraint (Doostmohammadian et al., 2021, Doostmohammadian, 2023). In weight-balanced directed-graph variants, conservation is obtained from the Laplacian identity $V=\{1,\dots,n\}$7 rather than pairwise symmetry (Doostmohammadian et al., 2022).

A common misconception is that “anytime-feasible” means that all constraints are enforced exactly at every iterate. In most DanyRA formulations, the invariant applies to the shared coupling equality or inequality, while local box constraints are handled through penalties or barriers and therefore may be enforced only approximately unless an explicit projection mechanism is added. This distinction is stated directly in several papers and is central to interpreting the guarantee correctly (Doostmohammadian et al., 21 Oct 2025, Doostmohammadian et al., 2021).

3. Algorithmic mechanisms and nonlinear models

The principal DanyRA line is primal rather than dual: it drives gradient consensus while staying on the feasible manifold. For equality-constrained problems, the KKT conditions require primal feasibility and stationarity in the form $V=\{1,\dots,n\}$8 for all $V=\{1,\dots,n\}$9. DanyRA dynamics therefore couple gradient differences through Laplacian-like operators until $\mathcal{G}(k)=(V,E(k))$0, which recovers the stationarity condition without introducing a central coordinator or projection onto the global hyperplane (Doostmohammadian et al., 21 Oct 2025).

A distinguishing technical feature is explicit accommodation of sector-bounded nonlinearities. In the resilient 2025 formulation, $\mathcal{G}(k)=(V,E(k))$1 and $\mathcal{G}(k)=(V,E(k))$2 satisfy

$\mathcal{G}(k)=(V,E(k))$3

and are odd, sign-preserving, and monotone non-decreasing. The paper gives log-scale quantization and saturation/clipping as concrete examples, with the sector bounds entering the stability margins and step-size restrictions directly (Doostmohammadian et al., 21 Oct 2025). The same modeling device appears in earlier work on nonlinear agents, CPU scheduling under quantization, and energy resource management, where quantizers, saturation maps, and sign-based nonlinearities are treated as part of the algorithm rather than as exogenous perturbations (Doostmohammadian et al., 2021, Doostmohammadian et al., 2022, Doostmohammadian, 2023).

Acceleration has been pursued along two main routes. One is signum-based gain shaping, in which $\mathcal{G}(k)=(V,E(k))$4 with $\mathcal{G}(k)=(V,E(k))$5 accelerates convergence near the optimum and $\mathcal{G}(k)=(V,E(k))$6 with $\mathcal{G}(k)=(V,E(k))$7 accelerates convergence far from it. The resulting algorithm remains all-time feasible because the signum powers are odd and appear only in antisymmetric pairwise exchanges (Doostmohammadian et al., 2024). The second route is momentum. The 2025 momentum-based algorithm introduces $\mathcal{G}(k)=(V,E(k))$8 and augments the update with $\mathcal{G}(k)=(V,E(k))$9, while preserving feasibility through symmetric weight-balanced mixing and matched delayed differences (Doostmohammadian et al., 8 Mar 2025).

A more recent development departs from pure Laplacian dynamics and adopts a control-theoretic synthesis. In the inequality-coupled setting, a safe feedback controller based on control barrier functions enforces a discrete-time contraction of the global residual, while a per-agent projection onto an affine admissible set corrects a nominal primal-dual step. This construction yields anytime feasibility for coupled inequalities and extends to equality constraints with a linear convergence theorem (Wu et al., 4 Aug 2025). This suggests that DanyRA is best understood as an invariant-preserving design principle rather than a single update formula.

4. Connectivity, delays, failures, and robustness

DanyRA methods are designed for networks that are not continuously reliable. A central relaxation is uniform connectivity: there exists $W(k)$0 such that the union graph

$W(k)$1

is connected for all $W(k)$2. This is weaker than persistent connectivity and permits temporary disconnections, topology switching, and intermittent loss of communication links (Doostmohammadian et al., 21 Oct 2025, Doostmohammadian et al., 2024).

The resilient 2025 formulation extends the invariant and convergence analysis to heterogeneous bounded delays, packet drops, and substantial link failures. Delays are modeled by $W(k)$3, with time stamps and aggregation of past gradient differences. The same antisymmetry argument used in the delay-free case preserves $W(k)$4 under the delayed update. Convergence then requires a smaller admissible step-size, scaling inversely with $W(k)$5 (Doostmohammadian et al., 21 Oct 2025). Related delay-tolerant constructions appear in distributed energy management and momentum-based scheduling, where identical delays are used on both sides of a gradient difference to preserve the invariant exactly (Doostmohammadian, 2023, Doostmohammadian et al., 8 Mar 2025).

Link failures are treated particularly explicitly in the percolation-based analysis of the 2025 resilient algorithm. If each instantaneous link fails independently with probability $W(k)$6, then an Erdős–Rényi graph may be disconnected at any given time, but the union graph remains connected with probability $W(k)$7 when $W(k)$8, where $W(k)$9 is the bond-percolation threshold given in the paper. Under this condition, DanyRA converges with probability $i$0 despite severe intermittent disconnections (Doostmohammadian et al., 21 Oct 2025). The paper’s simulations include supercritical failure regimes such as $i$1 and $i$2, where feasibility remains exact and convergence is recovered by enlarging the connectivity window $i$3.

Robustness does not eliminate all limitations. If uniform connectivity fails over the required window, the invariant generally survives but optimality guarantees can break, and the iterate sequence may stall away from the optimizer. Likewise, tighter quantization or heavier saturation widens the sector bounds, forcing smaller step-sizes and slower convergence (Doostmohammadian et al., 21 Oct 2025). In the barrier-function-based inequality-coupled formulation, robustness to exogenous violations is instead mediated by a virtual queue with minimum buffer, which restores feasibility before pre-defined deadlines at the price of an explicit convergence-accuracy trade-off (Wu et al., 4 Aug 2025).

5. Convergence theory, rates, and complexity

Theoretical analysis across DanyRA variants centers on three statements: feasibility invariance, equilibrium characterization, and convergence. In the resilient primal method, any equilibrium satisfies $i$4, and convergence over uniformly-connected networks is guaranteed when

$i$5

With bounded delays, the denominator becomes $i$6, making the delay–stepsize trade-off explicit (Doostmohammadian et al., 21 Oct 2025). The proof uses the residual Lyapunov function $i$7, the $i$8-smoothness bound, and spectral inequalities for the Laplacian of the union graph.

Rate guarantees vary by formulation. The signum-accelerated continuous-time method proves global asymptotic convergence under uniform connectivity and identifies the linear case $i$9 with $f_i$0 decay, while emphasizing that the accelerated nonlinear rates are faster qualitatively but not given in closed form (Doostmohammadian et al., 2024). The control-barrier-function variant for equality-coupled problems goes further and establishes a linear convergence rate, while the inequality-coupled version converges exactly when the robustness buffer is zero and to an $f_i$1-neighborhood otherwise (Wu et al., 4 Aug 2025).

Per-iteration complexity is intentionally lightweight. In the resilient 2025 algorithm, each node computes a local gradient, applies $f_i$2 and $f_i$3, and aggregates neighbor contributions in $f_i$4 arithmetic; with delays, the per-iteration cost becomes $f_i$5. Memory is $f_i$6, and communication is one gradient message per neighbor per iteration, with time stamps when delays are present (Doostmohammadian et al., 21 Oct 2025). Earlier signum-based and quantized methods have similarly sparse per-node state, typically consisting of the primal variable, current gradient, and a small number of nonlinear or momentum auxiliaries (Doostmohammadian et al., 2024, Doostmohammadian et al., 8 Mar 2025).

Parameter tuning is correspondingly spectral. The admissible step-size scales positively with $f_i$7 and $f_i$8, and negatively with $f_i$9, $\sum_i x_i=b$0, and the connectivity or delay window (Doostmohammadian et al., 21 Oct 2025). The practical implication is that weaker uniform connectivity, coarser quantization, stronger clipping, or larger delay bounds all push DanyRA toward more conservative steps. In the barrier-function-based robust formulation, $\sum_i x_i=b$1 controls feasibility contraction speed, while the buffer $\sum_i x_i=b$2 trades off violation recovery against steady-state optimality error (Wu et al., 4 Aug 2025).

6. Variants, applications, and broader research context

The DanyRA label is explicit only in papers, but the underlying invariant has been developed across several related algorithmic families. Representative instances are summarized below.

Paper	Mechanism	Distinctive guarantee
“1st-Order Dynamics on Nonlinear Agents for Resource Allocation over Uniformly-Connected Networks” (Doostmohammadian et al., 2021)	First-order nonlinear consensus-based primal dynamics	Anytime feasibility under broad odd sign-preserving nonlinearities
“Distributed CPU Scheduling Subject to Nonlinear Constraints” (Doostmohammadian et al., 2022)	Nonlinear gradient-Laplacian protocols on undirected and weight-balanced directed graphs	Anytime feasibility with quantization and $\sum_i x_i=b$3-accuracy under uniform quantization
“Accelerated Distributed Allocation” (Doostmohammadian et al., 2024)	Signum-based accelerated primal dynamics	All-time feasibility with tunable $\sum_i x_i=b$4 acceleration
“Momentum-based Distributed Resource Scheduling Optimization Subject to Sector-Bound Nonlinearity and Latency” (Doostmohammadian et al., 8 Mar 2025)	Momentum-augmented feasible gradient-tracking	All-time feasibility with sector-bounded links and bounded delays
“Distributed Constraint-coupled Resource Allocation: Anytime Feasibility and Violation Robustness” (Wu et al., 4 Aug 2025)	CBF-based safe primal-dual controller with virtual queue	Anytime feasibility for coupled inequalities and linear convergence for equalities
“Distributed Allocation and Resource Scheduling Algorithms Resilient to Link Failure” (Doostmohammadian et al., 21 Oct 2025)	Primal Laplacian-coupled DanyRA with percolation-based resilience analysis	Exact feasibility under delays, packet drops, intermittent disconnections, and link failures

The application range is correspondingly broad. Equality-coupled DanyRA formulations have been used for economic dispatch, generator scheduling, and distributed energy resource management, where stopping the algorithm early without violating supply–demand balance is operationally important (Doostmohammadian, 2023, Doostmohammadian et al., 21 Oct 2025). CPU scheduling in data centers is another recurring benchmark, with local box constraints modeling server capacities and quantization modeling communication or implementation limits (Doostmohammadian et al., 2022, Doostmohammadian et al., 8 Mar 2025). Networked coverage control appears in the accelerated allocation paper as a canonical distributed allocation setting (Doostmohammadian et al., 2024).

A methodologically distinct but conceptually aligned branch appears in distributed SDN control. FD-ADMM for fair bandwidth allocation continuously outputs a feasible per-flow allocation $\sum_i x_i=b$5 at each iteration by combining per-link projections with a conservative min-aggregation across links, and this feasible sequence converges to the $\sum_i x_i=b$6-fair optimum (Allybokus et al., 2017, Allybokus et al., 2018). This line does not use Laplacian gradient consensus, but it realizes the same anytime-feasible principle.

The main point of comparison throughout the literature is with primal-dual and ADMM-style methods that enforce the coupling constraint only asymptotically. DanyRA formulations are repeatedly presented as structurally different in that feasibility is not an asymptotic residual but an invariant of the dynamics. That distinction is decisive in applications where overallocation, oversubscription, or imbalance cannot be tolerated even transiently, or where the algorithm may need to terminate before full convergence (Doostmohammadian et al., 2024, Doostmohammadian et al., 21 Oct 2025).