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QoS-based Beamforming and Compression Design for Cooperative Cellular Networks via Lagrangian Duality

Xilai Fan, Ya-Feng Liu, Liang Liu, Tsung-Hui Chang

TL;DR

There is no duality gap between the considered joint optimization problem and its Lagrangian dual by showing the tightness of its semidefinite relaxation (SDR) and an efficient algorithm is proposed based on the above duality result for solving the considered problem.

Abstract

This paper considers the quality-of-service (QoS)-based joint beamforming and compression design problem in the downlink cooperative cellular network, where multiple relay-like base stations (BSs), connected to the central processor via rate-limited fronthaul links, cooperatively transmit messages to the users. The problem of interest is formulated as the minimization of the total transmit power of the BSs, subject to all users' signal-to-interference-plus-noise ratio (SINR) constraints and all BSs' fronthaul rate constraints. In this paper, we first show that there is no duality gap between the considered joint optimization problem and its Lagrangian dual by showing the tightness of its semidefinite relaxation (SDR). Then, we propose an efficient algorithm based on the above duality result for solving the considered problem. The proposed algorithm judiciously exploits the special structure of an enhanced Karush-Kuhn-Tucker (KKT) conditions of the considered problem and finds the solution that satisfies the enhanced KKT conditions via two fixed point iterations. Two key features of the proposed algorithm are: (1) it is able to detect whether the considered problem is feasible or not and find its globally optimal solution when it is feasible; (2) it is highly efficient because both of the fixed point iterations in the proposed algorithm are linearly convergent and evaluating the functions in the fixed point iterations are computationally cheap. Numerical results show the global optimality and efficiency of the proposed algorithm.

QoS-based Beamforming and Compression Design for Cooperative Cellular Networks via Lagrangian Duality

TL;DR

There is no duality gap between the considered joint optimization problem and its Lagrangian dual by showing the tightness of its semidefinite relaxation (SDR) and an efficient algorithm is proposed based on the above duality result for solving the considered problem.

Abstract

This paper considers the quality-of-service (QoS)-based joint beamforming and compression design problem in the downlink cooperative cellular network, where multiple relay-like base stations (BSs), connected to the central processor via rate-limited fronthaul links, cooperatively transmit messages to the users. The problem of interest is formulated as the minimization of the total transmit power of the BSs, subject to all users' signal-to-interference-plus-noise ratio (SINR) constraints and all BSs' fronthaul rate constraints. In this paper, we first show that there is no duality gap between the considered joint optimization problem and its Lagrangian dual by showing the tightness of its semidefinite relaxation (SDR). Then, we propose an efficient algorithm based on the above duality result for solving the considered problem. The proposed algorithm judiciously exploits the special structure of an enhanced Karush-Kuhn-Tucker (KKT) conditions of the considered problem and finds the solution that satisfies the enhanced KKT conditions via two fixed point iterations. Two key features of the proposed algorithm are: (1) it is able to detect whether the considered problem is feasible or not and find its globally optimal solution when it is feasible; (2) it is highly efficient because both of the fixed point iterations in the proposed algorithm are linearly convergent and evaluating the functions in the fixed point iterations are computationally cheap. Numerical results show the global optimality and efficiency of the proposed algorithm.
Paper Structure (39 sections, 10 theorems, 85 equations, 7 figures, 1 table, 1 algorithm)

This paper contains 39 sections, 10 theorems, 85 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Prior Works
  3. Our Contributions
  4. Organization
  5. System Model and Problem Formulation
  6. System Model
  7. Problem Formulation
  8. SDR of (P) and Its Tightness
  9. Proposed Fixed Point Iteration Algorithm
  10. Enhanced KKT Conditions
  11. Solving for the Dual Variables
  12. Solving Eqs. \ref{['equ:dual_const2']} and \ref{['equ:dual_var2']} for $\left\{\boldsymbol{\mathbf{\Lambda}}_m^{ }\right\}$
  13. Solving Eqs. \ref{['equ:dual_var1']} and \ref{['equ:dual_const1']} for $\boldsymbol{\mathbf{\beta}}$
  14. Dual Fixed Point Iteration
  15. Solving for the Primal Variables
  16. ...and 24 more sections

Key Result

Theorem 1

Suppose that problem SDR is strictly feasible, and let $(\{\boldsymbol{\mathbf{V}}_k^{ }\}, \boldsymbol{\mathbf{Q}})$ be its solution. Then $\boldsymbol{\mathbf{V}}_k$ is of rank one for all $k\in\mathcal{K}$.

Figures (7)

  • Figure 1: The flow chart of the proposed algorithm for solving the enhanced KKT conditions.
  • Figure 2: An illustration of solving \ref{['equ:equ_Bm']} for $\boldsymbol{\mathbf{Q}}$.
  • Figure 3: (a) The convergence rate of the dual fixed point iteration \ref{['equ:dual_iter']} with $\bar{\gamma} = 4$ dB; (b) The convergence rate of the primal fixed point iteration \ref{['equ:pri_iter']} with $\bar{\gamma} = 4$ dB.
  • Figure 4: (a) The dual/primal objective values versus the iteration number for feasible problem instances with different SINR targets $\bar{\gamma}$; (b) The dual objective values versus the iteration number for infeasible problem instances with different SINR targets $\bar{\gamma}$.
  • Figure 5: The average objective value (i.e., the total transmit power) of different algorithms with different system parameters $(M, K, \bar{\gamma}, \bar{C})$.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Proposition 1
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6