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Joint Optimization of Radio and Computational Resources for Multicell Mobile-Edge Computing

Stefania Sardellitti, Gesualdo Scutari, Sergio Barbarossa

arXiv:1412.8416v1cs.NIcs.IT

TL;DR

The paper addresses energy-efficient computation offloading in a MIMO multicell system where intercell interference and latency couple radio transmission with shared cloud computation. It jointly optimizes transmit covariance matrices and CPU allocations, deriving a closed-form single-user solution and SCA-based multiuser algorithms. Numerical results show that the proposed algorithms outperform disjoint optimization schemes.

  • Problem

    Dense multicell offloading requires jointly handling radio resources, shared cloud computation, intercell interference, and latency while minimizing mobile-user energy consumption.

  • Method

    The paper derives a closed-form global solution for the single-user case and develops SCA algorithms for the multiuser nonconvex problem, including centralized and distributed implementations.

  • Results

    The proposed algorithms outperform disjoint optimization schemes and converge to local optimal solutions of the multiuser nonconvex problem.

  • Takeaways & Limitations

    Joint radio and computational resource optimization is especially beneficial for applications with high computational load and few bits exchanged for program migration.

Abstract

from arXiv · show

Migrating computational intensive tasks from mobile devices to more resourceful cloud servers is a promising technique to increase the computational capacity of mobile devices while saving their battery energy. In this paper, we consider a MIMO multicell system where multiple mobile users (MUs) ask for computation offloading to a common cloud server. We formulate the offloading problem as the joint optimization of the radio resources-the transmit precoding matrices of the MUs-and the computational resources-the CPU cycles/second assigned by the cloud to each MU-in order to minimize the overall users' energy consumption, while meeting latency constraints. The resulting optimization problem is nonconvex (in the objective function and constraints). Nevertheless, in the single-user case, we are able to express the global optimal solution in closed form. In the more challenging multiuser scenario, we propose an iterative algorithm, based on a novel successive convex approximation technique, converging to a local optimal solution of the original nonconvex problem. Then, we reformulate the algorithm in a distributed and parallel implementation across the radio access points, requiring only a limited coordination/signaling with the cloud. Numerical results show that the proposed schemes outperform disjoint optimization algorithms.

I. INTRODUCTION

The introduction motivates mobile-edge computation offloading as a response to limited mobile battery capacity and presents a dense multicell problem that jointly optimizes radio and cloud computational resources under latency and power constraints.

  • I. INTRODUCTION: Limited battery lifetime threatens the deployment of computation-intensive applications on increasingly connected mobile devices.The introduction also notes growing mobile traffic and heterogeneous IoT devices with varying computational capabilities.
  • I. INTRODUCTION: Mobile Cloud Computing enables mobile users to access virtualized cloud resources on demand for computation offloading.Cloudlets and small-cell base stations bring radio access and computational resources closer to mobile users, while MEC targets proximity, low latency, and high-rate access.
  • I. INTRODUCTION: Dense small-cell MEC introduces intercell interference, making offloading more challenging than the special cases studied previously.The paper addresses this setting through a common cloud connected to multiple SCeNBs, with users in different cells potentially interfering.
  • I. INTRODUCTION: The offloading problem minimizes mobile-terminal energy consumption while satisfying transmit-power and latency constraints.Its variables include each MIMO user’s transmit covariance or precoding matrix and the cloud CPU cycles/second assigned to each user.
  • I. INTRODUCTION: Users may compute locally or offload to the cloud, with offloading latency comprising input transmission, server execution, and output transmission.The radio variables are transmit covariance matrices subject to power-budget constraints, while CPU allocations are nonnegative and share the total computational budget.
  • I. INTRODUCTION: Latency couples communication and computation, linking wireless transmission, remote execution, backhaul transfer, and result delivery.The formulation jointly optimizes radio covariance matrices and computational-rate allocations under a shared cloud CPU budget.

III. THE SINGLE-USER CASE

In the single-user, interference-free case, the offloading problem admits a globally optimal closed-form solution. Offloading is feasible under a latency condition, and the optimal radio strategy has a water-filling-like structure coupled to computational resources.

  • The single-user offloading problem minimizes mobile-user energy subject to latency and transmit-power constraints.The formulation includes the cloud CPU-resource limit as well as the radio power budget.
  • Offloading is feasible if the wired-network delay is below the maximum tolerable delay and full wireless and computational utilization can satisfy the latency constraint.The condition requires ˜T > 0, with r(Q) = rmax and f = fT in the limiting case.
  • The nonconvex single-user problem can be converted into a convex equivalent problem whose global optimum is available in closed form.The auxiliary problem Qs is convex, and Problems Ps and Qs are equivalent under strict feasibility.
  • At the optimum, the latency constraint is met with equality, so energy minimization coincides with minimizing the mobile user's transmit power.This result follows from the unique solution characterized for the equivalent single-user problems.
  • The optimal covariance has a water-filling-like structure aligned with the equivalent-channel eigenvectors, but its water level depends on communication and computational parameters.Unlike classical water-filling, using the full power budget is generally not optimal because α is selected to satisfy latency with equality.

IV. COMPUTATION OFFLOADING OVER MULTIPLE-CELLS

The multi-cell offloading problem minimizes users’ sum energy under power, latency, and shared-cloud CPU constraints, but is nonconvex. The paper constructs strongly convex approximations that preserve first-order behavior and support distributed optimization.

  • The multi-cell formulation minimizes the sum of users’ energies subject to power budgets, latency constraints, and limited cloud computational resources.
  • The problem is nonconvex because both the objective function and latency constraints are nonconvex.
  • The SCA method replaces the original problem with a sequence of strongly convex problems using convex approximations around the current iterate.
  • The energy approximation convexifies the product of transmit power and transmission latency, which forms each user’s energy term.
  • The energy approximant preserves first-order behavior, satisfies the required approximation properties, and is strongly convex and separable in users’ variables.

2) Inner convexification of the constraints gin(Q, fin):

The constraint approximation uses the rate functions’ concave-convex structure to construct an inner convex approximation that preserves feasibility for the original problem.

  • The method builds an inner convex approximation of each nonconvex constraint around the current feasible iterate.
  • Because the approximated constraint upper-bounds the original constraint, any point satisfying the approximation remains feasible for the original problem.
  • The approximation retains the convex part of the reformulated constraint and linearizes its concave rate term.

3) Inner SCA algorithm: centralized implementation:

The centralized inner SCA algorithm repeatedly solves strongly convex approximations from a feasible starting point and updates the iterate with a step size. Its limit points are stationary solutions of the original problem.

  • Starting from a feasible Z0, the algorithm repeatedly solves a strongly convex subproblem, updates the iterate, and checks an energy-based termination criterion.
  • Every limit point of the generated iterates is a stationary solution of the original nonconvex problem, and none is a local maximum of energy.
  • The convergence theorem permits flexibility in selecting the regularization constant and step-size sequence, subject to its stated conditions.
  • A centralized implementation can run in the cloud, which collects system parameters, solves the convex subproblems, and sends solutions back to the small-cell base stations.
  • The proposed approximation and SCA algorithm address the absence of an additively separable convex/nonconvex decomposition in the sum-energy objective.

V. DISTRIBUTED IMPLEMENTATION

The distributed implementation chooses approximations that make the convexified subproblems decomposable across small-cell base stations, enabling parallel solution with limited cloud signaling.

  • The distributed algorithms are designed to converge to local optimal solutions while reducing communication overhead from the centralized implementation.
  • Separable approximations allow the convexified problems to decompose into smaller subproblems solved in parallel across the small-cell base stations.
  • Different valid constraint approximants trade off convergence speed, complexity, communication overhead, and required system-parameter knowledge.
  • Preserving the convex component of the original constraint can prevent decomposition across small-cell base stations because of nonadditive coupling among covariance variables.
  • The first distributed candidate exploits Lipschitz continuity of rate-function gradients, while an alternative approximation is also constructed to achieve separability.

1) Per-cell optimization via dual decomposition:

The paper develops dual-decomposition procedures that solve convexified offloading subproblems in parallel across SCeNBs, with convergence guarantees and an alternative slack-variable reformulation that avoids Lipschitz-constant knowledge.

  • Per-cell optimization via dual decomposition: Dualizing side constraints decomposes each convexified subproblem across SCeNBs, enabling parallel per-cell strongly convex minimizations.The dual problem has zero duality gap, and each SCeNB computes its local minimizer for a shared multiplier.
  • Per-cell optimization via dual decomposition: Algorithm 3 updates the master-node multipliers using diminishing step sizes after parallel SCeNB computations.The procedure alternates local solves, multiplier updates, and termination checks.
  • Per-cell optimization via dual decomposition: The multiplier sequence converges to a solution of the dual problem, and the associated primal sequence converges to the unique solution of the convexified subproblem.The stated conditions require positive step sizes with vanishing magnitude, divergent sum, and square-summable squares.
  • Alternative decomposition via slack variables: Introducing slack variables yields an equivalent formulation whose stationary solutions can be computed through a sequence of strongly convex problems.The reformulation preserves stationary solutions of the original problem in both directions.
  • Alternative decomposition via slack variables: The alternative decomposition avoids requiring Lipschitz constants, retains convergence under the stated theorem conditions, and admits a closed-form local solution for one subproblem.Second-order dual methods are reported to significantly enhance practical convergence speed.
  • Alternative decomposition via slack variables: The slack-variable subproblems also decouple across SCeNBs in the dual domain, and their local solutions can be computed in parallel.The partial Lagrangian has an additive structure across cells, while the reformulated problem remains connected to the original through stationarity equivalence.

VI. NUMERICAL RESULTS

Numerical experiments evaluate joint radio-computational optimization against disjoint allocation and examine convergence behavior. Joint optimization performs better in low-η applications, while the proposed algorithm converges in few iterations and is insensitive to initialization in the reported tests.

  • VI. NUMERICAL RESULTS: The experiments use a two-cell MIMO network with four active users per cell and two antennas at each transceiver.Unless otherwise stated, the simulations use fT = 2 · 10^7, ˜T = 0.1, w = 10^5, and snr = 10dB.
  • Joint vs. disjoint optimization: Joint optimization yields a considerable gain over disjoint optimization for low η, corresponding to applications transferring many bits for a given computational load.The comparison uses Algorithm 2 against the Disjoint Resource Allocation algorithm.
  • Joint vs. disjoint optimization: Overall energy consumption decreases for computationally intensive applications characterized by high η.Here η = w_in/b_in measures computational load transferred per enabling bit.
  • On the convergence speed: The proposed algorithm converges in very few iterations across different latency limits and receive-antenna counts.The plotted energy is averaged over 100 independent channel realizations.
  • On the convergence speed: With 1,000 independent initializations, the algorithm repeatedly reached practically the same result.This test evaluates sensitivity to local minima in the nonconvex optimization problem.

VII. CONCLUSIONS

The paper formulates multicell computation offloading as joint radio and computational resource optimization under latency and power constraints. It provides a closed-form single-user solution, convergent centralized and distributed multiuser algorithms, and numerical evidence favoring joint over disjoint optimization.

  • VII. CONCLUSIONS: Dense radio-access deployments provide proximity access to computation but introduce intercell interference, motivating joint resource optimization.The target is minimizing mobile users’ energy consumption while satisfying latency and power-budget constraints.
  • VII. CONCLUSIONS: In the single-user case, the nonconvex problem has a global optimum available in closed form.The result applies to the interference-free single-user setting.
  • VII. CONCLUSIONS: For the multi-cell multiuser case, centralized and distributed SCA algorithms converge provably to local optima.The distributed implementation is designed for the nonconvex joint optimization problem.
  • VII. CONCLUSIONS: Numerical results show that the proposed algorithms outperform disjoint optimization schemes.The reported conclusion also identifies high computational load and few exchanged bits as favorable offloading conditions.

A. Proof of Theorem 1

The proof establishes the single-user theorem by relating stationary points of the nonconvex formulation to solutions of a convex reformulation and by proving pseudo-convexity of the energy objective.

  • Proof of Theorem 1: The proof reduces global optimality to showing that the energy objective is pseudo-convex on the relevant convex feasible set.A stationary point of a pseudo-convex objective over a convex set is globally optimal.
  • Proof of Theorem 1: Every stationary point of the convex problem Qs is also a stationary point of the original problem Ps, and conversely.The equivalence is established through corresponding KKT conditions and multiplier constructions.
  • Proof of Theorem 1: The reverse implication reconstructs the KKT system of Qs from a stationary point of Ps and establishes the necessary equality condition.The proof then obtains the single-user solution expression and f⋆ = fT.
  • Proof of Theorem 1: The final algebra checks the remaining convex-problem conditions and computes the auxiliary quantities through the procedure described in Algorithm 1.The proof uses the rank of Q⋆ and Slater’s condition in this construction.
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