DAUR Algorithm: Blockchain Metaverse Optimization

• DAUR algorithm is a joint optimization framework that balances user association, offloading, and resource allocation in blockchain-enabled Metaverse wireless networks.
• It integrates local and server-side processing with blockchain functions, addressing coupled delay and energy consumption via fractional programming and semidefinite relaxation.
• The method employs alternating optimization and auxiliary-variable transformations to manage nonconvexity, ensuring a stationary solution for maximizing data processing efficiency.

Searching arXiv for the specified DAUR papers and closely related versions to ground the article in the cited literature. The DAUR algorithm—DPE-Aware User Association and Resource Allocation—is a joint optimization framework for blockchain-enabled Metaverse wireless communication systems that seeks to maximize data processing efficiency (DPE), defined as processed data bits divided by the sum of delay and energy consumption. It is formulated for systems in which mobile or VR users partially process tasks locally and partially offload them to edge Metaverse servers that also execute blockchain functions such as block generation, propagation, and validation. In this setting, DAUR addresses the coupled allocation of communication and computation resources, user–server association, and task partitioning under resource constraints and blockchain-induced overheads. The principal formulations and algorithmic structure are presented in "Data Processing Efficiency Aware User Association and Resource Allocation in Blockchain Enabled Metaverse over Wireless Communications" (Qian et al., 2024) and its later version "Enhancing Data Processing Efficiency in Blockchain Enabled Metaverse over Wireless Communications" (Qian et al., 7 Jul 2025).

1. Problem setting and performance objective

DAUR is designed for a blockchain-enabled Metaverse over wireless architecture comprising a set of users $\mathcal{N}=\{1,\dots,N\}$ and a set of Metaverse servers $\mathcal{M}=\{1,\dots,M\}$. Each user has a computation-intensive NFT or Metaverse task of size $d_n$ bits. A task can be split between local execution and server-side execution: the locally processed portion is $(1-\varphi_n)d_n$, while the offloaded portion is $\varphi_n d_n$, where $\varphi_n\in[0,1]$ is the offloading ratio (Qian et al., 2024).

The servers are modeled as joint MEC/Metaverse nodes and blockchain participants. They receive user data via wireless FDMA uplink, allocate CPU to offloaded Metaverse processing, and also perform blockchain-related operations associated with the processed data. These blockchain operations include block generation, block propagation, and validation, and they introduce additional delay and CPU consumption beyond the standard uplink-plus-edge-computing pipeline (Qian et al., 7 Jul 2025).

The central metric is Data Processing Efficiency (DPE): $$\text{DPE} = \frac{\text{processed data bits}}{\text{delay} + \text{energy}}.$$ The denominator is modeled more precisely as a weighted sum of delay and energy, with weights $\omega_t$ and $\omega_e$. On the user side, the cost is

$$\text{cost}^{(u)}_n = \omega_t T^{(up)}_n + \omega_e E^{(up)}_n,$$

and on the server side,

$\mathcal{M}=\{1,\dots,M\}$0

The overall objective is a sum of DPE ratios across local and offloaded processing terms, weighted by user- and pair-specific preference coefficients $\mathcal{M}=\{1,\dots,M\}$1 and $\mathcal{M}=\{1,\dots,M\}$2 (Qian et al., 2024).

This formulation treats throughput, latency, and energy as a unified optimization target. The papers explicitly motivate this by noting that immersive Metaverse applications require low latency, mobile devices have limited battery and CPU, and blockchain functionality introduces extra computation and propagation delay (Qian et al., 7 Jul 2025).

2. System model and resource-coupling structure

The user side is parameterized by task size $\mathcal{M}=\{1,\dots,M\}$3, maximum CPU capacity $\mathcal{M}=\{1,\dots,M\}$4, and maximum transmit power $\mathcal{M}=\{1,\dots,M\}$5. Local CPU usage is controlled by $\mathcal{M}=\{1,\dots,M\}$6, so the CPU assigned to local processing is $\mathcal{M}=\{1,\dots,M\}$7. The corresponding local processing delay and energy are

$\mathcal{M}=\{1,\dots,M\}$8

Here $\mathcal{M}=\{1,\dots,M\}$9 denotes CPU cycles per bit and $d_n$0 is the effective switched capacitance (Qian et al., 2024).

Wireless transmission uses FDMA, so no inter-user interference is assumed. If $d_n$1 is the fraction of server $d_n$2's bandwidth $d_n$3 allocated to user $d_n$4, and $d_n$5 is the fraction of the maximum user power $d_n$6, then the uplink rate is

$d_n$7

subject to

$d_n$8

The transmission delay and energy for offloaded bits are

$d_n$9

The association variable $(1-\varphi_n)d_n$0 satisfies $(1-\varphi_n)d_n$1, so each user is associated with exactly one server (Qian et al., 7 Jul 2025).

At the server side, each node $(1-\varphi_n)d_n$2 has CPU capacity $(1-\varphi_n)d_n$3 and total wireless bandwidth $(1-\varphi_n)d_n$4. A fraction $(1-\varphi_n)d_n$5 of server $(1-\varphi_n)d_n$6's CPU is assigned to user $(1-\varphi_n)d_n$7, with

$(1-\varphi_n)d_n$8

This allocation is further split by $(1-\varphi_n)d_n$9 into a Metaverse data-processing share $\varphi_n d_n$0 and a blockchain share $\varphi_n d_n$1. The server-side data-processing and block-generation terms are

$\varphi_n d_n$2

$\varphi_n d_n$3

$\varphi_n d_n$4

$\varphi_n d_n$5

Blockchain propagation and validation are represented by

$\varphi_n d_n$6

The server interconnection rate is $\varphi_n d_n$7 (Qian et al., 2024).

These definitions create the structural coupling that motivates DAUR. Server selection, offloading, uplink allocation, user power, local CPU usage, server CPU slicing, and blockchain processing shares are all interdependent. A server with strong radio conditions but high CPU load may degrade DPE, and conversely a lightly loaded server may be unattractive if its channel is poor or if propagation and validation overhead dominate (Qian et al., 7 Jul 2025).

3. Original optimization problem and sources of nonconvexity

The original optimization problem, denoted $\varphi_n d_n$8, maximizes the sum of local and offloaded DPE ratios over the decision variables

$\varphi_n d_n$9

Its objective is

$\varphi_n\in[0,1]$0

The constraint set includes one-server-per-user association, offloading bounds, wireless bandwidth constraints, server CPU allocation constraints, and user power/CPU usage bounds (Qian et al., 2024).

The papers identify several distinct sources of difficulty. First, the objective is a sum of ratios, which is nonconvex. Second, the denominator terms are nonlinear in the optimization variables and include quadratic CPU-energy expressions. Third, the uplink rate

$\varphi_n\in[0,1]$1

is itself nonconvex in $\varphi_n\in[0,1]$2. Fourth, the association variables are binary, and products such as $\varphi_n\in[0,1]$3 and $\varphi_n\in[0,1]$4 create a mixed-integer nonlinear structure. The resulting problem is described as highly nonconvex and NP-hard (Qian et al., 7 Jul 2025).

A common misconception would be to interpret DAUR as merely a resource allocator for an edge network with a blockchain label attached. The model used in the cited works explicitly includes blockchain-specific block generation, propagation, and validation delays and CPU costs in the server-side denominator of DPE. The algorithm is therefore constructed around a cost function that differs materially from standard MEC formulations (Qian et al., 2024).

4. Transformations underlying DAUR

DAUR proceeds by converting $\varphi_n\in[0,1]$5 into a sequence of problems that are tractable under alternating optimization. The first step introduces auxiliary variables $\varphi_n\in[0,1]$6, $\varphi_n\in[0,1]$7, $\varphi_n\in[0,1]$8, and $\varphi_n\in[0,1]$9 to convert ratio terms and delay expressions into inequality-constrained forms, producing an equivalent summation problem $$\text{DPE} = \frac{\text{processed data bits}}{\text{delay} + \text{energy}}.$$0 (Qian et al., 7 Jul 2025).

The second step introduces Lagrange multiplier-type parameters $$\text{DPE} = \frac{\text{processed data bits}}{\text{delay} + \text{energy}}.$$1 and $$\text{DPE} = \frac{\text{processed data bits}}{\text{delay} + \text{energy}}.$$2, yielding problem $$\text{DPE} = \frac{\text{processed data bits}}{\text{delay} + \text{energy}}.$$3 with objective

$$\text{DPE} = \frac{\text{processed data bits}}{\text{delay} + \text{energy}}.$$4

At a KKT point,

$$\text{DPE} = \frac{\text{processed data bits}}{\text{delay} + \text{energy}}.$$5

with analogous expressions on the server side. The papers state that this recovers the original ratios and establishes equivalence to $$\text{DPE} = \frac{\text{processed data bits}}{\text{delay} + \text{energy}}.$$6 at KKT points (Qian et al., 2024).

DAUR then partitions the variables into two blocks:

• Block 1: $$\text{DPE} = \frac{\text{processed data bits}}{\text{delay} + \text{energy}}.$$7
• Block 2: $$\text{DPE} = \frac{\text{processed data bits}}{\text{delay} + \text{energy}}.$$8

For Block 2, the troublesome term $$\text{DPE} = \frac{\text{processed data bits}}{\text{delay} + \text{energy}}.$$9 is handled by introducing an auxiliary variable $\omega_t$0 and applying a quadratic transform or related fractional-programming device, rewriting the term in a form that is concave in the resource-allocation variables when $\omega_t$1 is fixed. This yields Problem $\omega_t$2, which is solved by an inner loop alternating between optimization over $\omega_t$3 and updating $\omega_t$4 through a closed-form relation (Qian et al., 7 Jul 2025).

For Block 1, the authors use an analytical result that under $\omega_t$5 the optimal CPU partition ratio is

$\omega_t$6

which removes $\omega_t$7 from subsequent optimization. The remaining terms are converted into a QCQP through variable stacking into a vector $\omega_t$8. This QCQP is then lifted to an SDR by defining a PSD matrix

$\omega_t$9

so quadratic forms become trace terms. In the journal-style presentation (Qian et al., 7 Jul 2025), the rank-one requirement on $\omega_e$0 is addressed through a difference-of-convex (DC) penalty

$\omega_e$1

with linearization of the spectral norm term around the previous iterate. In the earlier version (Qian et al., 2024), the SDR solution is followed by a projection of the relaxed association variables to a feasible discrete assignment using the Hungarian algorithm.

This sequence of transformations is the distinguishing algorithmic content of DAUR. It combines sum-of-ratios reformulation, alternating optimization, fractional programming, QCQP/SDR lifting, and either DC-based rank-one enforcement or SDR-plus-rounding, depending on the version considered (Qian et al., 2024, Qian et al., 7 Jul 2025).

5. Algorithmic procedure

DAUR is presented as an outer iterative scheme with two inner subprocedures. The inputs include channel gains, CPU capacities, power limits, bandwidths, blockchain parameters, weights $\omega_e$2, preference parameters $\omega_e$3, and convergence tolerances $\omega_e$4 (Qian et al., 7 Jul 2025).

Initialization sets

$\omega_e$5

with an initial association matrix $\omega_e$6 chosen as a simple mapping such as round-robin. The initial auxiliary variables $\omega_e$7 and $\omega_e$8 are then computed using their closed-form KKT relationships (Qian et al., 7 Jul 2025).

The outer loop proceeds as follows.

First, the algorithm runs the resource-allocation FP loop for fixed association and offloading decisions. At FP iteration $\omega_e$9, it solves the concave problem $$\text{cost}^{(u)}_n = \omega_t T^{(up)}_n + \omega_e E^{(up)}_n,$$0 for $$\text{cost}^{(u)}_n = \omega_t T^{(up)}_n + \omega_e E^{(up)}_n,$$1 using fixed $$\text{cost}^{(u)}_n = \omega_t T^{(up)}_n + \omega_e E^{(up)}_n,$$2, then updates $$\text{cost}^{(u)}_n = \omega_t T^{(up)}_n + \omega_e E^{(up)}_n,$$3 via

$$\text{cost}^{(u)}_n = \omega_t T^{(up)}_n + \omega_e E^{(up)}_n,$$4

in the later exposition (Qian et al., 7 Jul 2025), or equivalently through the stated closed-form update in the earlier formulation (Qian et al., 2024). The loop terminates when the relative improvement in the $$\text{cost}^{(u)}_n = \omega_t T^{(up)}_n + \omega_e E^{(up)}_n,$$5 objective is below $$\text{cost}^{(u)}_n = \omega_t T^{(up)}_n + \omega_e E^{(up)}_n,$$6.

Second, the algorithm runs the association/offloading optimization step for fixed communication and compute allocations. With $$\text{cost}^{(u)}_n = \omega_t T^{(up)}_n + \omega_e E^{(up)}_n,$$7, the problem is turned into a QCQP and then relaxed to an SDR. In (Qian et al., 7 Jul 2025), the relaxed solution is iteratively refined through the DC penalty method to approximately satisfy rank one, after which a rank-one approximation of $$\text{cost}^{(u)}_n = \omega_t T^{(up)}_n + \omega_e E^{(up)}_n,$$8 is extracted and rounded to a binary association satisfying $$\text{cost}^{(u)}_n = \omega_t T^{(up)}_n + \omega_e E^{(up)}_n,$$9. The paper reports comparison among several rounding approaches—Hungarian, randomized, secondary discrete problem, and greedy—and states that rank‑1 approximation yields the best DPE with moderate computation. In (Qian et al., 2024), the continuous assignment is projected with the Hungarian algorithm to obtain a one-server-per-user discrete solution.

After both blocks have been updated, DAUR refreshes $\mathcal{M}=\{1,\dots,M\}$00 and $\mathcal{M}=\{1,\dots,M\}$01 using the KKT-based closed forms and checks the outer stopping rule: $\mathcal{M}=\{1,\dots,M\}$02 The final output is

$\mathcal{M}=\{1,\dots,M\}$03

Both versions describe the outcome as a stationary point of the transformed problem, with correspondence to the original DPE maximization through the reformulation and KKT relations (Qian et al., 2024, Qian et al., 7 Jul 2025).

6. Complexity, convergence, and numerical behavior

The reported worst-case complexity for the FP block is

$\mathcal{M}=\{1,\dots,M\}$04

and the QCQP/SDR block has a similar order with accuracy parameter $\mathcal{M}=\{1,\dots,M\}$05. The association reconstruction step is reported as either Hungarian algorithm or rank-1 approximation via SVD, with complexity $\mathcal{M}=\{1,\dots,M\}$06. With $\mathcal{M}=\{1,\dots,M\}$07 outer iterations, the total complexity is given as

$\mathcal{M}=\{1,\dots,M\}$08

in the later presentation (Qian et al., 7 Jul 2025).

The convergence claims are local rather than global. The cited works state that the individual transformation steps preserve equivalence at KKT points or yield tight relaxations; that the Block 2 subproblem is concave for fixed auxiliary variables; and that the DC procedure converges to a critical or stationary point of the penalized rank-one problem. The overall algorithm is therefore described as converging to a stationary point rather than a global optimum (Qian et al., 7 Jul 2025).

The numerical setup includes a circular cell of radius $\mathcal{M}=\{1,\dots,M\}$09 m, random user and server placement, path loss $\mathcal{M}=\{1,\dots,M\}$10, Rayleigh fading, noise PSD $\mathcal{M}=\{1,\dots,M\}$11 dBm, server bandwidth $\mathcal{M}=\{1,\dots,M\}$12 MHz, user maximum power $\mathcal{M}=\{1,\dots,M\}$13 W, user CPU $\mathcal{M}=\{1,\dots,M\}$14 GHz, server CPU $\mathcal{M}=\{1,\dots,M\}$15 GHz, $\mathcal{M}=\{1,\dots,M\}$16, $\mathcal{M}=\{1,\dots,M\}$17, $\mathcal{M}=\{1,\dots,M\}$18, task sizes between $\mathcal{M}=\{1,\dots,M\}$19 KB and $\mathcal{M}=\{1,\dots,M\}$20 KB, block size $\mathcal{M}=\{1,\dots,M\}$21 MB, wired rate $\mathcal{M}=\{1,\dots,M\}$22 Mbps, $\mathcal{M}=\{1,\dots,M\}$23, and balanced weights $\mathcal{M}=\{1,\dots,M\}$24 (Qian et al., 7 Jul 2025).

The papers compare DAUR against four baselines: RUCAA, GUCAA, AAUCO, and GUCRO. Reported DPE values for the default $\mathcal{M}=\{1,\dots,M\}$25 case differ slightly between the two versions:

Method	Reported DPE in (Qian et al., 2024)	Reported DPE in (Qian et al., 7 Jul 2025)
RUCAA	80.78	80.78
GUCAA	80.38	80.38
AAUCO	81.87	83.25
GUCRO	84.82	84.82
DAUR	86.48	87.87

The later version explicitly attributes the difference to a refined SDP/DC approach, stating that it improves DPE and reduces QCQP iterations relative to the conference version (Qian et al., 7 Jul 2025).

Across sensitivity studies, DPE increases with server bandwidth, server CPU, user CPU, and user power, while DAUR remains the best-performing method among those compared. The reported delay–energy weight studies and preference-weight studies show that DPE is sensitive to the normalization induced by $\mathcal{M}=\{1,\dots,M\}$26 and scales with the preference coefficients $\mathcal{M}=\{1,\dots,M\}$27. This suggests that the absolute magnitude of DPE should be interpreted together with the chosen weighting and scaling parameters, not in isolation (Qian et al., 2024).

7. Assumptions, limitations, and interpretation

The stated assumptions are strong and largely centralized. DAUR presumes a controller with access to global or system-wide information such as user locations, channel gains, CPU capacities, and other resource states. Channel conditions and task arrivals are treated as static or quasi-static over the optimization horizon. The wireless model uses FDMA with no inter-user interference. The blockchain model is simplified to single-hop server interconnection with propagation and validation delays aggregated at a coarse level; detailed consensus behavior such as PoW, PoS, or leader-election dynamics is not modeled (Qian et al., 2024).

These assumptions delimit the scope of the algorithm. The cited works explicitly note several limitations: centralized optimization may create signaling burden and privacy concerns; computational complexity grows rapidly with system size; SDR and rounding or DC-based approximate rank-one recovery do not furnish global optimality; and time-varying channels, stochastic arrivals, mobility, and richer blockchain semantics are excluded (Qian et al., 7 Jul 2025).

The papers also identify extensions including decentralized or distributed DAUR, dynamic or online DAUR, richer blockchain modeling, utility-aware DPE formulations, learning-based policy approximation, and tighter polynomial-optimization refinements (Qian et al., 7 Jul 2025). This suggests a broader interpretation of DAUR not as a single immutable solver, but as a methodological template for jointly optimizing offloading, association, and blockchain-aware resource allocation under DPE-style objectives.

Within the cited literature, DAUR is therefore best understood as a structured response to a very specific optimization problem: maximizing a bits-per-weighted-delay-and-energy objective in a blockchain-enabled Metaverse edge network where communication, computation, and blockchain overhead are explicitly coupled. Its technical significance lies in the way it decomposes a mixed-integer sum-of-ratios program into tractable subproblems through auxiliary-variable reformulation, alternating optimization, fractional programming, and semidefinite methods (Qian et al., 2024, Qian et al., 7 Jul 2025).