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Multi-Objective Resource Allocation for Secure Communication in Cognitive Radio Networks with Wireless Information and Power Transfer

Derrick Wing Kwan Ng, Ernest S. Lo, Robert Schober

arXiv:1403.0054v2cs.IT

TL;DR

The paper addresses how to allocate resources in cognitive-radio SWIPT networks when secure communication, energy harvesting, transmit power, and interference leakage impose competing objectives. It proposes a weighted-Tchebycheff Pareto framework, convexifies the problem through SDP relaxation, and reports near-optimal suboptimal schemes alongside the resulting objective trade-offs.

  • Problem

    Single-objective formulations cannot capture the trade-offs among transmit power minimization, energy harvesting efficiency maximization, and interference leakage minimization in secure CR-SWIPT networks.

  • Method

    A weighted-Tchebycheff multi-objective formulation uses SDP relaxation and primal-dual SDP solutions to construct Pareto-optimal resource allocations under imperfect eavesdropper CSI.

  • Results

    The proposed schemes achieve close-to-optimal performance, while simulations reveal trade-offs among transmit power, energy harvesting efficiency, and interference leakage.

  • Takeaways & Limitations

    Transmit-power minimization generally also yields low interference leakage, whereas maximizing energy harvesting efficiency requires higher transmit power and interference leakage.

Abstract

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In this paper, we study resource allocation for multiuser multiple-input single-output secondary communication systems with multiple system design objectives. We consider cognitive radio networks where the secondary receivers are able to harvest energy from the radio frequency when they are idle. The secondary system provides simultaneous wireless power and secure information transfer to the secondary receivers. We propose a multi-objective optimization framework for the design of a Pareto optimal resource allocation algorithm based on the weighted Tchebycheff approach. In particular, the algorithm design incorporates three important system objectives: total transmit power minimization, energy harvesting efficiency maximization, and interference power leakage-to-transmit power ratio minimization. The proposed framework takes into account a quality of service requirement regarding communication secrecy in the secondary system and the imperfection of the channel state information of potential eavesdroppers (idle secondary receivers and primary receivers) at the secondary transmitter. The adopted multi-objective optimization problem is non-convex and is recast as a convex optimization problem via semidefinite programming (SDP) relaxation. It is shown that the global optimal solution of the original problem can be constructed by exploiting both the primal and the dual optimal solutions of the SDP relaxed problem. Besides, two suboptimal resource allocation schemes for the case when the solution of the dual problem is unavailable for constructing the optimal solution are proposed. Numerical results not only demonstrate the close-to-optimal performance of the proposed suboptimal schemes, but also unveil an interesting trade-off between the considered conflicting system design objectives.

I. INTRODUCTION

The paper formulates Pareto-optimal resource allocation for secure SWIPT in cognitive radio networks, jointly addressing transmit power, energy harvesting, and interference leakage objectives under imperfect eavesdropper CSI.

  • Existing formulations focus on single objectives and cannot study trade-offs among transmit power, energy harvesting efficiency, and interference leakage-to-transmit power ratio.
  • The proposed non-convex multi-objective formulation jointly minimizes transmit power and interference leakage-to-transmit power ratio while maximizing energy harvesting efficiency.It accounts for imperfect CSI of idle secondary receivers and primary receivers, producing Pareto-optimal resource allocation policies.
  • SDP relaxation recasts the non-convex problem as a convex optimization problem, while primal and dual optimal solutions enable construction of the original problem's global optimum.
  • Two suboptimal resource allocation schemes address cases where the dual SDP solution is unavailable for constructing the optimal solution.
  • Energy harvesting efficiency maximization conflicts with transmit power minimization, while maximum harvesting efficiency requires high interference leakage and high transmit power.
  • The secondary transmitter concurrently sends information and artificial noise, using artificial noise both to secure communication and facilitate energy transfer.The system serves one active receiver with information and idle receivers with energy transfer in each scheduling slot.

III. RESOURCE ALLOCATION PROBLEM FORMULATION

This section defines secrecy, energy-transfer, and primary-protection QoS measures for a cognitive-radio SWIPT network, then formulates resource-allocation objectives using channel and system models.

  • III. RESOURCE ALLOCATION PROBLEM FORMULATION: The resource-allocation formulation defines QoS measures for wireless power transfer, secure communication, and protection of primary receivers.
  • III. RESOURCE ALLOCATION PROBLEM FORMULATION: The formulation assumes identical noise characteristics for all primary receivers because of similar hardware architectures.
  • III. RESOURCE ALLOCATION PROBLEM FORMULATION: The notation introduces H = hhH, Gk = gkgH_k, and Lj = ljlH_j to represent channel-related Hermitian matrices.
  • A. System Achievable Rate and Secrecy Rate: With perfect receiver CSI, the achievable rate is defined from the desired secondary receiver's received SINR Γ.
  • A. System Achievable Rate and Secrecy Rate: Achievable rates are also defined for idle secondary receivers and primary receivers, which are treated as potential eavesdroppers.
  • A. System Achievable Rate and Secrecy Rate: The secrecy rate quantifies the maximum reliable secret-information rate when eavesdroppers cannot decode the received signal.

B. Energy Harvesting Efficiency

The secondary receivers harvest RF energy while idle, and energy harvesting efficiency is defined relative to the secondary transmitter’s radiated power. Interference leakage to primary receivers is also treated as a resource-allocation measure.

  • Idle secondary receivers harvest RF energy to extend their lifetimes.
  • Energy harvesting efficiency is the ratio of total power harvested by idle secondary receivers to total power radiated by the secondary transmitter.
  • Each idle receiver’s conversion efficiency ηk lies between 0 and 1, while primary-transmitter and AWGN power are neglected for worst-case robust design.
  • Because primary and secondary receivers share spectrum, interference leakage to primary receivers must be incorporated into secondary resource allocation.

D. Channel State Information

The framework models secondary- and primary-receiver channels with estimates plus bounded uncertainties, reflecting limited interaction and channel variation. These models support robust secrecy, energy-harvesting, and interference-aware resource allocation.

  • D. Channel State Information: The system uses TDD with slowly time-varying channels and obtains secondary-receiver downlink CSI through reciprocal uplink training.
  • D. Channel State Information: Idle secondary-receiver channels are modeled as gk = ĝk + ∆gk, where ∆gk belongs to a bounded uncertainty region Ωk.
  • D. Channel State Information: Primary-receiver channels use the analogous model lj = l̂j + ∆lj because primary receivers may not interact continuously with the secondary transmitter.
  • E. Optimization Problem Formulations: The multi-objective design jointly considers energy-harvesting efficiency, total transmit power, and interference leakage-to-transmit-power ratio under secure-communication constraints.
  • E. Optimization Problem Formulations: The secrecy constraints use required and tolerable SINR parameters, with the secrecy-rate lower bound determined by their selected values.
  • E. Optimization Problem Formulations: Weighted objective coefficients generate Pareto-optimal resource-allocation policies and generalize the three single-objective formulations.

IV. SOLUTION OF THE OPTIMIZATION PROBLEMS

The optimization problems are non-convex because of selected objectives and a secrecy constraint. The paper applies semidefinite programming relaxation to obtain tractable convex formulations.

  • The formulations are non-convex in the optimization variables because of objective functions 1 and 3 and constraint C1.
  • Semidefinite programming relaxation recasts Problems 1–4 as convex optimization problems for tractable solution and relaxation-tightness analysis.

A. Semidefinite Programming Relaxation

The formulation transforms the multi-objective resource-allocation problems into equivalent finite convex forms, then applies SDP relaxation to address the remaining rank constraint. Under the stated feasibility assumption, the relaxed problem supports recovery of globally optimal solutions using primal and dual information.

  • The energy-harvesting, transmit-power, interference-leakage, and multi-objective formulations are rewritten using new optimization variables W, V, and ξ.
  • The transformed problems are equivalent to the original formulations, and their original solutions can be recovered from transformed-problem solutions using (23).
  • The analysis assumes feasibility because unfavorable channels or overly stringent QoS requirements can make the considered problems infeasible.Multiple optimal solutions may exist, and the proposed scheme finds at least one global optimum.
  • Auxiliary variables decouple nested semi-infinite constraints and enable their replacement by a finite collection of convex constraints and linear matrix inequalities.The S-Procedure is applied to robustify the channel-dependent constraints.
  • After relaxing Rank(W) = 1, the remaining problem is a convex SDP solvable by numerical optimization tools; a rank-one relaxed solution is already optimal for the original problem.Without rank one, the relaxed objective provides a performance upper bound for the original problem.
  • The global optimum can be recovered from the SDP primal solution together with the dual solution, while two suboptimal schemes avoid requiring the dual solution.

B. Optimality Condition for SDP Relaxation

The SDP relaxation is tight: a rank-one beamforming solution with the same objective value can be constructed from the relaxed primal and dual solutions. This establishes global optimality for the original problems and supports single-objective recovery.

  • A rank-one feasible solution with the same objective value as the relaxed optimum exists and can be constructed using the relaxed primal and dual solutions.
  • Because an achievable optimal rank-one beamforming matrix always exists, the SDP relaxation still yields the global optimum of the original problem.
  • Each of the three single-objective problems is solved by applying the multi-objective problem with the corresponding weight set to one and the others to zero.
  • The stated computational complexity is polynomial in the numbers of secondary users, primary users, and transmit antennas for a given solution accuracy.A tailor-made interior-point method can further reduce computational complexity.

C. Suboptimal Resource Allocation Schemes

When the dual solution needed for exact rank-one recovery is unavailable, the paper proposes two suboptimal resource-allocation schemes based only on the primal relaxed solution.

  • The proposed suboptimal schemes use the primal solution of the SDP-relaxed problem when the dual Lagrange multiplier matrix is unavailable.

1) Suboptimal Resource Allocation Scheme 1:

Scheme 1 uses the relaxed SDP solution directly when it is rank one; otherwise, it constructs a rank-one beamformer from the dominant eigenvector and reoptimizes its power and other variables jointly.

  • 1) Suboptimal Resource Allocation Scheme 1:: Scheme 1 first solves the SDP-relaxed problem and obtains the global optimum whenever the resulting W has rank one.
  • 1) Suboptimal Resource Allocation Scheme 1:: The constants F ∗p are used in the multi-objective formulation to study trade-offs and recover single-objective solutions, but need not equal the optimal value of Problem p.
  • 1) Suboptimal Resource Allocation Scheme 1:: When W has higher rank, the scheme forms a rank-one beamforming matrix from its dominant eigenvector.The eigenvalues are ordered descending, and the first eigenvector is used to define the suboptimal beamforming vector.
  • 1) Suboptimal Resource Allocation Scheme 1:: A scalar variable Pb controls the power assigned to the constructed suboptimal beamforming matrix.
  • 1) Suboptimal Resource Allocation Scheme 1:: The resulting optimization problem is jointly convex in its optimization variables and can be solved using efficient numerical solvers.

2) Suboptimal Resource Allocation Scheme 2:

Scheme 2 uses Gaussian randomization to construct a rank-one beamforming solution when the SDP beamforming matrix has rank greater than one, allowing a performance–complexity trade-off. Simulations examine the proposed and baseline trade-off regions under two objective cases, showing that the baseline can approach optimal performance in high-power regimes while the objectives exhibit distinct conflicts and alignments.

  • Suboptimal resource allocation scheme 2: When Rank(W*) > 1, scheme 2 constructs a suboptimal rank-one beamforming matrix using Gaussian randomization.It samples r ∼ CN(0, I_NT), forms w_sub = UΣ^1/2r, and sets W_sub = P_bw_subw_sub^H.
  • Suboptimal resource allocation scheme 2: Scheme 2 re-optimizes Θ_sub for each randomized beamforming vector and can improve performance by selecting the best result across repeated trials.This flexibility is not offered by scheme 1 and trades computational complexity against system performance.
  • Trade-off regions: In Case I, transmit-power minimization and interference-leakage minimization are partially aligned, whereas harvested-power maximization conflicts with both.Maximizing harvested power requires full-power transmission, which also produces high average interference leakage.
  • Trade-off regions: In Case II, energy-harvesting efficiency and IPTR can remain favorable at high transmit power because both ratios are invariant to simultaneous positive scaling of W and V.The trade-off region includes points near 30 dBm average transmit power and 0.1% average IPTR.
  • Baseline comparison: The baseline scheme approaches the proposed optimal SDP-based trade-off region for energy-harvesting efficiency and total harvested power in the high-transmit-power regime.It uses MRT for information beamforming, exploits the same CSI, and reduces computational complexity by roughly half by not fully exploiting that CSI.

B. Average Total Harvested Power and Average Energy Harvesting Efficiency

The simulations characterize trade-offs among harvested power, energy harvesting efficiency, transmit power, interference leakage, and IPTR across resource allocation schemes and receiver counts. Proposed schemes outperform the baseline while preserving the secrecy-rate requirement under imperfect CSI.

  • Harvested power and energy harvesting efficiency increase with transmit power, conflicting with transmit-power minimization.
  • The proposed optimal scheme provides a larger trade-off region than the baseline, including 2.7 dB less transmit power in Figure 5(a).The two proposed suboptimal schemes perform close to the optimal SDP trade-off region.
  • Increasing the number of secondary receivers shifts trade-off curves upward and rightward because more idle receivers harvest power and become potential eavesdroppers.Additional potential eavesdroppers require more artificial noise to neutralize information leakage.
  • All proposed resource allocation schemes guarantee Csec ≥5.6582 bit/s/Hz despite imperfect channel state information.
  • Harvested power and energy harvesting efficiency increase with interference leakage and IPTR, respectively, revealing conflicts with minimizing those interference objectives.The proposed suboptimal schemes remain close to the optimal trade-off region, while the baseline achieves a smaller region.
  • Higher transmit power need not increase interference leakage or IPTR when multiple-antenna degrees of freedom are properly exploited.The baseline has 10 dB more interference leakage in Figure 7(a) than the proposed schemes.

APPENDIX

The appendix proves that the SDP-relaxed formulation is convex and that an optimal relaxed solution can be transformed into an equivalent rank-one beamforming solution. The construction preserves feasibility and objective value.

  • The Charnes-Cooper transformation converts the original formulation into an equivalent problem with normalized transmit covariance trace.The transformed constraint is Tr(W) + Tr(V) = 1, with ξ constrained nonnegative without loss of optimality.
  • A constructed solution preserves the optimal objective value and all constraints while producing a rank-one beamforming matrix.The scalars used in the construction are obtained by solving a resulting convex optimization problem.
  • The relaxed problem is jointly convex and satisfies Slater’s condition, making KKT conditions necessary and sufficient for optimality.
  • The dual formulation uses nonnegative scalar and positive-semidefinite matrix multipliers associated with secrecy, covariance, normalization, and receiver constraints.
  • Complementary slackness places the columns of the optimal beamforming matrix in the null space of its dual multiplier matrix.
  • The proof decomposes the optimal beamforming matrix into a positive semidefinite component and a rank-one component aligned with the relevant null-space structure.

C1: Tr(Hf W)

This appendix section lists the transformed constraints governing the constructed beamforming and energy-transfer variables.

  • The constructed variables satisfy positive semidefinite covariance constraints, nonnegative auxiliary variables, and the normalized trace equality.The listed constraints include C5–C7 and nonnegativity requirements for δk, γj, ϕk, and ωj.
  • Constraints C10 and C11 impose the corresponding matrix conditions on the constructed beamforming, energy-transfer, and auxiliary variables.
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