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Multi-Objective Optimization for Robust Power Efficient and Secure Full-Duplex Wireless Communication Systems

Yan Sun, Derrick Wing Kwan Ng, Jun Zhu, Robert Schober

arXiv:1509.01425v1cs.IT

TL;DR

Secure full-duplex multiuser communication requires power-efficient resource allocation that protects simultaneous downlink and uplink transmissions despite conflicting power objectives and imperfect channel information. The paper formulates this as a weighted-Tchebycheff multi-objective problem and solves it through semidefinite-programming relaxation. The resulting simulations reveal the downlink–uplink power trade-off, confirm robustness against eavesdroppers, and show that secure uplink transmission is enabled by the full-duplex base station.

  • Problem

    Secure full-duplex systems need to minimize total downlink and uplink transmit power while guaranteeing secure simultaneous transmission under imperfect channel information.

  • Method

    The paper uses a weighted-Tchebycheff multi-objective formulation and semidefinite-programming relaxation to obtain globally optimal secure resource-allocation solutions.

  • Results

    Simulations reveal the trade-off between total downlink and uplink transmit power, confirm robustness against eavesdroppers, and show that an FD base station enables secure uplink transmission.

  • Takeaways & Limitations

    Full-duplex operation provides a way to guarantee simultaneous downlink and uplink security while achieving power savings over a baseline scheme.

Abstract

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In this paper, we investigate the power efficient resource allocation algorithm design for secure multiuser wireless communication systems employing a full-duplex (FD) base station (BS) for serving multiple half-duplex (HD) downlink (DL) and uplink (UL) users simultaneously. We propose a multi-objective optimization framework to study two conflicting yet desirable design objectives, i.e., total DL transmit power minimization and total UL transmit power minimization. To this end, the weighed Tchebycheff method is adopted to formulate the resource allocation algorithm design as a multi-objective optimization problem (MOOP). The considered MOOP takes into account the quality-of-service (QoS) requirements of all legitimate users for guaranteeing secure DL and UL transmission in the presence of potential eavesdroppers. Thereby, secure UL transmission is enabled by the FD BS and would not be possible with an HD BS. The imperfectness of the channel state information of the eavesdropping channels and the inter-user interference channels is incorporated for robust resource allocation algorithm design. Although the considered MOOP is non-convex, we solve it optimally by semidefinite programming (SDP) relaxation. Simulation results not only unveil the trade-off between the total DL transmit power and the total UL transmit power, but also confirm the robustness of the proposed algorithm against potential eavesdroppers.

I. INTRODUCTION

The introduction motivates secure full-duplex communication as a response to spectral inefficiency, interference, and wireless security risks. It identifies an open problem in jointly minimizing downlink and uplink power under realistic channel uncertainty, then presents a robust multi-objective SDP-based solution.

  • Half-duplex systems separate uplink and downlink transmissions orthogonally, causing significant spectral-efficiency loss.
  • Full-duplex systems enable simultaneous downlink and uplink transmission but face residual self-interference and co-channel interference.
  • Prior secure-transmission studies do not directly address secure full-duplex multiuser MIMO under practical interference and channel-state-information conditions.
  • Secure full-duplex communication must protect both downlink and uplink users, whereas a half-duplex base station cannot secure uplink transmission by jamming eavesdroppers.
  • The paper formulates a multi-objective problem that jointly minimizes total downlink and uplink transmit power for secure full-duplex multiuser MIMO.
  • Imperfect eavesdropper and interference-channel information is incorporated, and semidefinite-programming relaxation yields Pareto-optimal resource-allocation policies.

II. SYSTEM MODEL

The paper models a full-duplex base station serving simultaneous half-duplex downlink and uplink users while accounting for roaming users that may eavesdrop. Artificial noise is transmitted alongside downlink information to interfere with potential eavesdroppers.

  • The system contains an FD BS, K legitimate HD downlink users, J legitimate HD uplink users, and M roaming users.
  • The FD BS uses multiple antennas to transmit downlink signals and receive uplink signals simultaneously in the same frequency band.
  • Roaming users have multiple antennas and are treated as potential eavesdroppers because they may intercept legitimate users’ information.
  • The downlink transmit vector combines K independent information streams with artificial noise modeled as z ∼ CN(0, Z).
  • The model includes downlink, uplink, self-interference, inter-user interference, eavesdropping, and additive-noise channels.

III. RESOURCE ALLOCATION PROBLEM FORMULATION

The formulation defines achievable and secrecy rates for simultaneous FD downlink and uplink transmission under interference and eavesdropping. Artificial noise enables the FD BS to protect both directions of communication.

  • III. RESOURCE ALLOCATION PROBLEM FORMULATION: The section defines performance metrics and formulates downlink and uplink transmit-power minimization problems.
  • A. Achievable Rate and Secrecy Rate: Downlink and uplink achievable rates are based on their received SINRs, with zero-forcing receive beamforming adopted for uplink detection.
  • A. Achievable Rate and Secrecy Rate: Zero-forcing beamforming is selected because it can approach MMSE beamforming when receiver noise is not dominant and reduces computational complexity.
  • A. Achievable Rate and Secrecy Rate: The model incorporates imperfect self-interference cancellation using a noise parameter 0 < ρ ≪ 1.
  • A. Achievable Rate and Secrecy Rate: Potential eavesdroppers are assumed to cancel multiuser interference before decoding desired downlink or uplink information.
  • A. Achievable Rate and Secrecy Rate: Unlike an HD BS, the FD BS can guarantee downlink and uplink security simultaneously through artificial-noise transmission.

B. Channel State Information

The paper distinguishes accurate transmission-link CSI from imperfect interference and eavesdropping-channel CSI. It models the latter uncertainties deterministically and incorporates them into robust resource-allocation formulations.

  • B. Channel State Information: CSI for uplink and downlink transmission links is assumed perfect over each transmission period.
  • B. Channel State Information: CSI for inter-user interference and eavesdropping channels is imperfect because the base station updates those estimates only at scheduling-slot boundaries.
  • B. Channel State Information: Channel uncertainties are modeled deterministically as bounded errors around available CSI estimates.
  • B. Channel State Information: The uncertainty sets contain all possible channel errors with bounds determined by channel coherence time and transmission duration.
  • III. RESOURCE ALLOCATION PROBLEM FORMULATION: The optimization considers downlink and uplink power objectives together because their minimizations conflict in secure FD communication.
  • III. RESOURCE ALLOCATION PROBLEM FORMULATION: The formulation imposes QoS and eavesdropper-rate constraints to guarantee secrecy when the robust optimization problem is feasible.
  • III. RESOURCE ALLOCATION PROBLEM FORMULATION: Higher downlink information and artificial-noise power increases self-interference, while higher uplink power increases downlink interference and eavesdropping risk.
  • III. RESOURCE ALLOCATION PROBLEM FORMULATION: Multi-objective optimization is used to study the trade-off between total downlink and total uplink transmit power.

IV. SOLUTION OF THE OPTIMIZATION PROBLEM

The non-convex robust optimization problems are transformed into convex semidefinite programs using LMI reformulations, slack variables, and rank relaxation. A theorem establishes that an optimal rank-one solution can be recovered under feasibility assumptions.

  • The original problems are non-convex because of constraints C1–C4 and infinitely many inequalities induced by channel uncertainty.
  • The resulting Problems 1, 2, and 3 are solved by semidefinite programming relaxation.
  • The equivalent Problem 3 uses an epigraph representation and an auxiliary variable to unify the optimization treatment.
  • The S-Procedure transforms robust quadratic constraints such as C1 into equivalent linear matrix inequalities.
  • Generalized S-Procedure transformations handle the uncertain constraints C3 and C4, while a slack Hermitian matrix addresses coupled estimation errors.
  • The rank-one constraint is removed to obtain a tractable convex SDP, although rank-constrained optimization is NP-hard.
  • If the relaxed solution has Rank(W_k) = 1, it is optimal for the original problem; otherwise, Theorem 1 guarantees construction of a rank-one optimal matrix under feasibility.
  • The optimal beamforming vector is recovered from the rank-one matrix using its principal eigenvector.

V. RESULTS

The simulations evaluate the proposed multi-objective resource allocation scheme under specified user, antenna, fading, and channel-estimation-error settings.

  • Simulation setup: The evaluation considers K = 3 downlink users, J = 7 uplink users, and M = 2 potential eavesdroppers.
  • Simulation setup: The full-duplex base station is cell-centered and equipped with NT antennas, while each eavesdropper has NR = 2 antennas.
  • Channel model: Downlink, uplink, inter-user-interference, and eavesdropping channels use independent identically distributed fading models, with the SI channel modeled separately.
  • Channel uncertainty and QoS: The simulations assume a 5 dB Rician factor, common maximum normalized estimation errors across relevant channels, and equal minimum SINRs within each user class.

A. Transmit Power Trade-off Region

The proposed scheme exposes a clear trade-off between total downlink and uplink transmit power and achieves a broader, more power-efficient trade-off region than the baseline.

  • Proposed trade-off: Total uplink transmit power decreases monotonically as total downlink transmit power increases, and minimizing one objective raises the other.
  • Antenna effects: Increasing the number of base-station antennas saves substantial transmit power, but channel hardening produces diminishing returns.
  • Baseline comparison: The proposed scheme is more power efficient than the baseline for both downlink and uplink transmission because it globally exploits available degrees of freedom.
  • Baseline comparison: With NT = 12, the baseline saves only 1 dB of uplink power for a 2 dB increase in total downlink power.
  • Additional baselines: The alternative isotropic-radiation and half-duplex-base-station baselines cannot satisfy the adopted QoS constraints, so their results are omitted.

B. Average Total Transmit Power versus Minimum Required SINR

The proposed scheme is evaluated as SINR requirements increase, showing rising power demands and stronger robustness than the baseline under imperfect channel state information.

  • Downlink SINR: Downlink power grows more rapidly than uplink power as the minimum required downlink SINR increases.
  • Baseline comparison: The proposed scheme provides substantial power savings over the baseline across the considered downlink and uplink SINR scenarios.
  • Robustness: At ΓDL req = 12 dB with NT = 10, outage probability is 0.5% for the proposed scheme versus 99.3% for the baseline.
  • Robustness: The results indicate greater robustness and reliability against imperfect channel state information than the baseline.
  • Uplink SINR: Both downlink and uplink transmit powers increase as the minimum required uplink SINR increases.
  • Uplink SINR: Higher uplink power increases cross-channel interference, requiring additional downlink information-signal and artificial-noise power to preserve QoS and security.

C. Average User Secrecy Rate versus Minimum Required SINR

The proposed scheme maintains required secrecy rates for both downlink and uplink users under imperfect CSI, while the baseline often achieves higher secrecy rates at substantially greater power and with feasibility limits.

  • Average DL secrecy rate increases with ΓDL req, whereas average UL secrecy rate depends only weakly on ΓDL req.
  • The proposed scheme fulfills the minimum required secrecy rate for both DL and UL users despite imperfect eavesdropping- and CCI-channel CSI.This confirms simultaneous security of both links under robust optimization.
  • The baseline scheme achieves higher secrecy rates than the proposed scheme but requires substantially larger DL and UL transmit powers.
  • The baseline scheme becomes infeasible when ΓDL req exceeds 12 dB and incurs high outage probability.
  • As ΓUL req increases, the proposed scheme continues to achieve secrecy rates above the minimum required user secrecy rate despite imperfect CSI.
  • The baseline scheme again uses significantly more DL and UL transmit power, has high outage probability, and becomes infeasible when ΓUL req exceeds 8 dB.

D. Average Transmit Power versus Maximum Channel Estimation Error

Average DL and UL transmit powers increase with channel-estimation error, while the proposed robust scheme retains feasibility and uses less power than the baseline.

  • Average total DL and UL transmit powers increase as the maximum normalized channel-estimation error increases.The FD BS must compensate for less accurate beam steering and interference suppression.
  • The baseline scheme consumes significantly higher power for both DL and UL transmission whenever the problem is feasible.
  • The proposed scheme remains robust to imperfect CSI while guaranteeing simultaneous secure DL and UL transmission.
  • The formulation jointly minimizes total DL and UL transmit powers through a non-convex MOOP solved using SDP relaxation.
  • The results reveal a trade-off between total DL and UL transmit power and show that FD operation can guarantee secure UL transmission unavailable with an HD BS.
  • The proposed scheme provides substantial power savings over the baseline scheme.

APPENDIX

The appendix develops determinant and trace inequalities used to relate the original secrecy constraints to equivalent matrix inequalities and establish lower bounds.

  • Applying the determinant-trace lemma yields implications connecting the secrecy constraints to trace and maximum-eigenvalue bounds.
  • Lemma 3 states that det(I + A) ≥ 1 + Tr(A) for positive semidefinite A, with equality exactly when Rank(A) ≤1.
  • The appendix uses det(I+AB) = det(I+BA) to establish equivalence between matrix expressions appearing in the proof.
  • Constraints C3 and C4 are rewritten into equivalent quadratic matrix inequalities involving the defined matrices S_j,m, M_j,m, and T_m.

C. Proof of Theorem 1

The proof shows that SDP relaxation preserves an optimal rank-one transmit-beamforming solution, including the boundary case where the UL objective receives zero weight.

  • The SDP-relaxed version of Problem 3 is jointly convex, enabling primal-dual analysis under Slater’s condition and strong duality.
  • KKT conditions are used to characterize the structure and rank of the optimal DL beamforming matrices W_k.
  • When λ1 = 0, the problem reduces to total UL transmit-power minimization, after which an auxiliary problem constructs a rank-one optimal solution.
  • Consequently, the globally optimal solution can always be obtained or constructed by solving at most two convex SDP optimization problems.
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