Intelligent Reflecting Surface Meets Mobile Edge Computing: Enhancing Wireless Communications for Computation Offloading

I. INTRODUCTION

• Edge computing moves computation toward mobile devices, while computation offloading compares local and edge execution using weighted time and energy utilities.Source: §I · p. 1
• IRSs reconfigure reflected signals to improve wireless propagation, especially when direct channels are weak because of obstacles, poor conditions, or absent line of sight.Source: §I · p. 1
• The paper formulates IRS-assisted offloading between mobile devices and an AP-hosted edge server, extending prior work with a comprehensive cost metric and customized rate constraints.Source: §I · pp. 1–2

II. RELATED WORK

• Prior studies separately examine edge offloading, IRS-assisted communications, or related uplink and downlink settings, while this work combines these topics.Source: §II · p. 2

III. SYSTEM MODEL

• The system contains a multi-antenna AP with an edge server and K single-antenna mobile devices, with IRS assistance and mutual interference during communication.Source: §III · p. 3

A. A Most Generic Model: Local Computing versus Edge Computing

• Each mobile device chooses local or edge execution by comparing utilities based on the time and energy required to complete its task.Source: A. A Most Generic Model: Local… · p. 3
• The IRS is represented by a diagonal phase-shift matrix whose reflecting amplitudes and angles configure the wireless environment and determine each device’s transmission rate.Source: A. A Most Generic Model: Local… · p. 4
• Edge execution includes uplink transmission, edge-server computation, mobile-device transmission and tail energy, and a weighted time–energy cost.Source: A. A Most Generic Model: Local… · pp. 3–4
• A device selects offloading when its gross edge-computing utility is at least its local-computing utility, with payments incorporated into the edge-computing utility.Source: A. A Most Generic Model: Local… · p. 4
• The edge server maximizes total payments subject to each device meeting its information-rate requirement, while payment expressions depend on task benefit and offloading cost.Source: A. A Most Generic Model: Local… · pp. 4–5

B. A Simplified Case

• When all devices choose edge computing, the offloading-dependent rates simplify to IRS-dependent rates R_k(φ), yielding Problem (P2).Source: B. A Simplified Case · p. 5
• Problem (P2) can be efficiently solved and therefore reduces the difficulty of solving the original Problem (P1).Source: B. A Simplified Case · p. 5
• Problem (P2) provides a lower bound for Problem (P1), while compensating devices can encourage future offloading and potentially increase long-term edge-server earnings.Source: B. A Simplified Case · p. 5

C. Signal Model

• The system uses a multi-antenna AP, K single-antenna devices, an IRS with N reflecting elements, direct and cascaded channels, and quasi-static flat fading.Source: C. Signal Model · p. 5
• The AP obtains channel-state information and adjusts IRS amplitudes and phases to optimize the edge-server objective.Source: C. Signal Model · p. 5
• The received signals include interference and receiver noise, and the SINR depends on the IRS configuration φ.Source: C. Signal Model · p. 5
• With unit bandwidth, each device’s information rate R_k(φ) is evaluated from its SINR using Shannon’s formula and an SINR-maximizing receiving vector.Source: C. Signal Model · pp. 5–6

D. Problem Formulation

• The paper states that its goal is to solve (P2) in (19), with Rk(φ) given by (27), and later discusses feasibility and solution of (P3).Source: D. Problem Formulation · p. 6
• Problem (P2) is reformulated as Problem (P3) by minimizing the sum of users’ rate-dependent terms with respect to the IRS variables.The offloading decisions are fixed, and R_k(φ) denotes each device’s information rate.Source: D. Problem Formulation · p. 6
• The feasibility and solution of (P3) are handled in separate subsequent procedures.Source: D. Problem Formulation · p. 6

V. FEASIBILITY PROBLEM

• The procedure provides a sufficient condition for feasibility and returns a feasible solution when the rate-constrained problem is identified as feasible.It addresses the difficulty of determining feasibility under arbitrary assigned rate requirements.Source: §V · p. 6
• The SINR constraints are reformulated through mean square error and an optimal linear receiver, enabling an equivalent feasibility formulation.The minimal MSE identity connects the receiver-based formulation to the original rate constraints.Source: §V · p. 6
• If the transformed problem's optimum is no greater than 1, the original problem is feasible; otherwise, the transformed criterion indicates infeasibility.The transformation exploits the quadratic, bi-convex structure of the MSE expression.Source: §V · pp. 6–7
• Block coordinate descent alternately updates the IRS variables and receiver-related variables while monotonically decreasing the auxiliary objective.The procedure stops with a feasible IRS configuration when the auxiliary value reaches at most 1.Source: §V · pp. 6–7
• Because the transformed problem is nonconvex and the algorithm obtains a suboptimal solution, finding a value no greater than 1 certifies feasibility but failure to do so does not establish infeasibility.The method's infeasibility conclusion is therefore not globally guaranteed.Source: §V · p. 7

VI. SOLVING THE OPTIMIZATION

• The algorithm tackles the sum-of-ratios objective by introducing equivalent transformations and auxiliary variables, assuming a feasible starting solution.The reformulated problem is handled through the auxiliary variables λ and µ.Source: §VI · p. 7
• The equivalent rate transformation yields convex subproblems that can be solved numerically within the alternating optimization procedure.The section also uses closed-form updates for selected variables while optimizing the IRS variables separately.Source: §VI · pp. 7–8
• The reformulated auxiliary problem preserves the original optimization through equivalent transformations rather than replacing it with an unrelated approximation.The derivation relies on the stated identities and parameterized reformulations.Source: §VI · p. 7
• Lemmas connect solutions of the auxiliary problem to the KKT conditions of the transformed optimization, enabling the original problem to be solved through the auxiliary formulation.The connection requires solving the auxiliary problem and satisfying the associated update conditions.Source: §VI · pp. 7–8
• The iterative algorithm alternates solving the auxiliary problem and updating λ and µ with a modified Newton method.The first step uses block coordinate descent because the auxiliary problem is jointly nonconvex in its variables.Source: §VI · p. 8
• The modified Newton residual converges to zero with global linear and local quadratic convergence rates.The resulting residual-based conditions provide a termination criterion for the iterative process.Source: §VI · p. 8

Simulation setup

• The simulation system uses one four-antenna AP, four single-antenna mobile users, one IRS, and N values of 30, 60, 90, and 120.The AP–IRS distance is 50 m, and the experiments use the stated fading, pathloss, noise, bandwidth, and transmit-power settings.Source: §VII · pp. 8–9, Figure 2 · p. 9

Feasibility-check convergence

• Algorithm 1 generally converges within 10 iterations, with the maximal feasible common rate requirement estimated between 2.3 and 2.5 Nats/s/channel-use.The experiment fixes one random channel realization, sets N=30, and varies the common requirement from 2.1 to 2.9 Nats/s/channel-use.Source: §VII · p. 9

IRS impact on feasibility

• Optimized IRS improves the feasibility probability by 40% across the tested rate-requirement range compared with no-IRS and random-IRS cases.The comparison uses 600 random channel realizations for each specified IRS size and common rate requirement.Source: §VII · pp. 9–10

Algorithm 2 convergence

• Algorithm 2 usually converges in several outer loops and exhibits superlinear convergence behavior, indicating fast convergence in the tested channel realizations.The experiments start Algorithm 2 from feasible points found for five random channel realizations.Source: §VII · p. 10, Figure 5 · p. 10, Figure 6 · p. 10

Optimizing (P3)

• Algorithm 2 effectively decreases the objective value of (P3) for various N, equivalently increasing the edge server’s earnings over random IRS elements.The comparison uses 10 random channel realizations and feasible solutions for each tested N.Source: §VII · p. 10, Figure 7 · p. 10

• The paper studies mobile edge computing aided by IRS and formulates edge-server earning maximization with information-rate constraints.Source: §VIII · p. 10
• An iterative algorithm checks rate-constraint feasibility and finds a feasible solution, while another algorithm optimizes the edge server’s earning from computation loading.Source: §VIII · p. 10
• Numerical results show that the IRS-aided scheme improves guaranteed information rates for mobile devices and increases edge-server earnings.Source: §VIII · p. 10

A. Proof of Lemma 1

• The proof of Lemma 1 introduces the KKT conditions of (P10) and identifies the associated Lagrangian multipliers.Source: A. Proof of Lemma 1 · p. 11
• Substituting the optimal variables and multipliers into selected KKT conditions of (P10) yields the results in (45) and (46).Source: A. Proof of Lemma 1 · p. 11
• The remaining KKT conditions of (P10) coincide with those of (P11) when its parameters are set to λ = λ⋆ and µ = µ⋆.Source: A. Proof of Lemma 1 · p. 11

B. Proof of Lemma 2

• For Lemma 2, an optimal solution of (P11) with λ = λ⋆ and µ = µ⋆ satisfies the KKT conditions associated with its constraints.Source: B. Proof of Lemma 2 · p. 11
• Together with the conditions in ②, these optimality conditions imply all KKT conditions of (P10), completing the proof of Lemma 2.Source: B. Proof of Lemma 2 · p. 11