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Matrix-Monotonic Optimization for MIMO Systems

Chengwen Xing, Shaodan Ma, Yiqing Zhou

arXiv:1312.1766v9cs.ITmath.IT

TL;DR

MIMO optimization often involves high-dimensional matrix variables, motivating reductions to vector-variable problems. The paper unifies such designs through matrix-monotonic optimization and derives optimal diagonalizable structures that simplify the problems.

  • Problem

    MIMO optimization involves high-dimensional matrix-variable problems, and deriving optimal matrix structures is a common route to reducing them to vector-variable problems.

    Source: §I · p. 2

  • Method

    The paper transforms selected MIMO matrix-variate optimization problems into a unified matrix-monotonic framework that exploits objective-function monotonicity to derive optimal solution structures.

    Source: §II · p. 6

  • Results

    The paper derives exact optimal diagonalizable structures for the considered matrix variables, significantly simplifying the investigated optimization problems.

    Source: §VII · p. 31

  • Takeaways & Limitations

    The framework addresses linear and nonlinear transceiver designs, training-sequence designs, and their robust counterparts, and extends to multiple matrix variables.

    Source: §VII · p. 31

    The analysis focuses mainly on cases where Ψ∝I or Σ∝I; the optimal solution structure remains difficult to derive when neither condition holds.

    Source: A. Matrix-Monotonic Optimization Problem · p. 13

Abstract

from arXiv · show

For MIMO systems, due to the deployment of multiple antennas at both the transmitter and the receiver, the design variables e.g., precoders, equalizers, training sequences, etc. are usually matrices. It is well known that matrix operations are usually more complicated compared to their vector counterparts. In order to overcome the high complexity resulting from matrix variables, in this paper we investigate a class of elegant multi-objective optimization problems, namely matrix-monotonic optimization problems (MMOPs). In our work, various representative MIMO optimization problems are unified into a framework of matrix-monotonic optimization, which includes linear transceiver design, nonlinear transceiver design, training sequence design, radar waveform optimization, the corresponding robust design and so on as its special cases. Then exploiting the framework of matrix-monotonic optimization the optimal structures of the considered matrix variables can be derived first. Based on the optimal structure, the matrix-variate optimization problems can be greatly simplified into the ones with only vector variables. In particular, the dimension of the new vector variable is equal to the minimum number of columns and rows of the original matrix variable. Finally, we also extend our work to some more general cases with multiple matrix variables.

I. INTRODUCTION · II. PROBLEM UNIFICATION

The paper uses matrix monotonicity to unify MIMO matrix-variable designs, derive optimal structures, and reduce the remaining optimization to vector variables. It applies this framework across multiple MIMO design settings and extends it to problems with multiple matrix variables.

I. INTRODUCTION

  • High-dimensional MIMO designs make optimal matrix-structure derivation central to reducing matrix-variable optimization to lower-dimensional problems with vector variables.The paper identifies transceiver, training-sequence, radar-waveform, and robust designs among the settings where optimal diagonalizable structures are sought.Source: §I · p. 2
  • The framework covers robust capacity and LMMSE transceiver designs, training-sequence design, dual-hop and multi-hop AF relaying, nonlinear DFE/THP designs, and multicarrier MIMO.The introduction also notes that training-sequence design is closely related to radar waveform optimization.Source: §I · pp. 3–4
  • The paper distinguishes its matrix-objective MMOPs from scalar covariance-objective optimization and targets closed-form optimal structures that simplify problems to vector-variable optimization.It also analyzes Pareto-optimal solutions, unitary-matrix transformations and inequalities, and more general problems with multiple matrix variables.Source: §I · pp. 4–5

II. PROBLEM UNIFICATION

  • For mutual-information designs, special cases of the general formulation include training design and linear transceiver design with perfect CSI; robust formulations incorporate channel-estimation effects.The paper also unifies additive and multiplicative Schur-convex/concave objective formulations for linear and nonlinear designs.Source: §II · pp. 6, 9
  • For dual-hop AF MIMO relaying without source precoding, the paper gives an exact optimal solution to the robust MSE design where prior work derived only a suboptimal solution under the stated matrix conditions.The comparison concerns the case Ψ̸ ∝ I and Σ ∝ I, as described in the passage.Source: §II · p. 9
  • The unified framework turns the considered MIMO matrix-variable designs into MMOPs, whose matrix objectives can be transformed into monotonic vector functions by choosing an optimal unitary matrix.The paper then treats the resulting problem as an MMOP; its objective is a decreasing vector function of the matrix eigenvalues.Source: §II · p. 10, F HFQ · p. 11

F HFQ

  • For the optimal Q, the unified matrix objective becomes a decreasing vector function of the eigenvalues, and the resulting optimization problem is treated as an MMOP.The optimal Q is derived case by case in a later section.Source: §II · p. 10, F HFQ · p. 11

III. FUNDAMENTALS OF MATRIX-MONOTONIC OPTIMIZATION · A. Matrix-Monotonic Optimization Problem

The section defines matrix-monotonic optimization as a multi-objective formulation with a positive-semidefinite matrix objective, then derives optimal structures under specified conditions. It also identifies limitations when neither Ψ nor Σ is proportional to the identity.

A. Matrix-Monotonic Optimization Problem

B. Optimal Solution of ΛF

After Theorem 1 fixes the optimal structure, the remaining variable is the rectangular diagonal matrix ΛF; its solutions are generally derived using KKT conditions and often take water-filling forms. The section uses multi-objective optimization to derive these solutions.

B. Optimal Solution of ΛF

  • The Pareto point set is preserved when the order constraint is removed, allowing the problem to be solved by scalarization.The order constraint can be realized through proper weighting, and the scalarization can efficiently achieve Pareto solutions.Source: B. Optimal Solution of ΛF · p. 16
  • A weighted single-objective scalarization of the Pareto solutions has a water-filling optimal solution.The Pareto solutions are attained by introducing positive weight factors and solving the resulting single-objective problem.Source: B. Optimal Solution of ΛF · pp. 16–17
  • The dual problem, which typically corresponds to QoS-based designs such as minimizing transmit power under a performance requirement, has the same solution as Theorem 1.Source: B. Optimal Solution of ΛF · p. 17

IV. UNITARY-MATRIX BASED TRANSFORMATIONS TO MMOPS · A. Basic Matrix Inequality Based Method

The optimal unitary matrix Q depends on the objective function; this section introduces matrix-inequality methods for deriving Q and applies four basic inequalities to optimal solutions across six cases.

IV. UNITARY-MATRIX BASED TRANSFORMATIONS TO MMOPS

  • The optimal unitary matrix Q depends on the objective function, and the paper introduces basic matrix-inequality and majorization-based methods for deriving it.Source: §IV · p. 17

A. Basic Matrix Inequality Based Method

B. Majorization Theory Based Methods · 1) Additive majorization theory based method: · 2) Multiplicative majorization theory based method:

The methods select unitary matrices according to whether the objective is Schur-concave or Schur-convex, with additive and multiplicative majorization applying to different diagonal quantities. The additive case also identifies the DFT matrix as a typical solution for an equal-diagonal condition.

1) Additive majorization theory based method:

B. Majorization Theory Based Methods

2) Multiplicative majorization theory based method:

V. EXTENSIONS TO MULTIPLE MATRIX-VARIATE CASES

The section extends matrix-monotonic optimization to problems with multiple matrix variables, including multi-hop AF MIMO relay transceiver design. It introduces unitary-matrix-based variables and develops conditions for simplifying and solving the resulting optimization.

V. EXTENSIONS TO MULTIPLE MATRIX-VARIATE CASES

  • The framework is extended from one matrix variable to multiple matrix variables, with multi-hop AF MIMO relaying as an example.The section takes up this more general case after the preceding single-matrix-variable treatment.Source: §V · p. 21

XK HKXK

  • Introducing unitary matrices Qk and new variables reformulates the multiple-matrix optimization problem.The supplied passage does not specify the variable definitions or the reformulated problem’s full expression.Source: XK HKXK · p. 21

FK HKFKQK

  • The reformulated problem decouples into K subproblems, each equivalent to MVOP 2, and yields a stated optimal structure when Ψk or Σk is proportional to I.When neither Ψk nor Σk is proportional to I, the stated structure is also a meaningful suboptimal solution; the optimal Qk values depend on the problem formulation.Source: FK HKFKQK · p. 22

A. Serial Structure · KMKQK · · · M1Q1] · B. Parallel Structure

The serial- and parallel-structure cases extend matrix-monotonic optimization to robust multi-hop relaying and MIMO-OFDM designs. The serial results derive optimal unitary matrices using majorization, while the parallel problem decouples into single-matrix subproblems; the treatment is limited to single-user cases.

A. Serial Structure

  • Linear transceiver designs have lower complexity, while nonlinear designs usually achieve better BER for a given modulation scheme.These are the two transceiver-design types considered for the serial structure.Source: A. Serial Structure · p. 23
  • Robust linear transceiver design for multi-hop AF MIMO relaying is formulated for additive Schur-convex or Schur-concave objectives.The paper presents this as one representative serial-structure example.Source: A. Serial Structure · p. 23

KMKQK · · · M1Q1]

  • For multi-hop AF MIMO relaying, THP precoding and DFE equalization support robust designs with multiplicative Schur-convex or Schur-concave objectives.The nonlinear precoder and equalizer are used at the source and destination, respectively, to enhance signal detection performance.Source: KMKQK · · · M1Q1] · pp. 23–24
  • For Case 9 and Case 10, the optimal unitary matrices Qk are derived using majorization theory.The derivation relies on an inequality for compatible rectangular matrices, with dimension defined by the minimum row and column counts and σi(Z) denoting the ith-largest singular value.Source: KMKQK · · · M1Q1] · p. 24
  • For additive and multiplicative Schur-concave objectives, Q1,opt diagonalizes the cited matrix expression,; additive Schur-convex objectives require equal diagonal elements of the cited expression.For multiplicative Schur-concave objectives, Q1,opt makes L have equal diagonal elements.Source: KMKQK · · · M1Q1] · p. 24

B. Parallel Structure

  • With parallel structure, the objective is a sum of matrix-monotone functions, and the optimization decouples into subproblems matching the single matrix-variate case.The same results as in the single matrix-variate case therefore apply; the formulation corresponds to MIMO-OFDM transceiver design.Source: B. Parallel Structure · p. 25
  • The multiple-matrix-variable derivation for the optimal diagonal matrices ΛFk follows the same logic as the single-matrix case.The paper considers single-user cases; the results do not simply extend to multi-user cases with mutual interference,, which are left for future research.Source: B. Parallel Structure · pp. 25–26

VI. SIMULATION RESULTS AND DISCUSSIONS · VII. CONCLUSIONS

Simulations show that proposed robust designs perform similarly across bounds and can outperform a non-robust design, while the unified matrix-monotonic framework derives simplifying matrix structures and extends to multiple matrix variables.

VI. SIMULATION RESULTS AND DISCUSSIONS

  • When transmitter and receiver correlation and estimation error are small, upper- and lower-bound gaps are very small; the gaps grow slightly as these factors increase.The bounds are assessed for the case Ψ̸∝I and Σ̸∝I; the derived lower-bound solution is suboptimal in that case.Source: §VI · pp. 26–27, Figure 2 · p. 27
  • The upper-bound, lower-bound, and iterative designs achieve exactly the same real-data detection MSE, even with high correlation and large channel errors.The bound curves themselves are not the real data detection MSEs; the comparison uses those MSEs.Source: §VI · p. 27, Figure 3 · p. 28
  • The proposed closed-form solution requires 11 dominant matrix operations, while the iterative algorithm uses 10 per iteration and takes 20–50 iterations in simulations.For an N×N matrix, each counted inverse, multiplication, or decomposition has complexity O(N^3); the iterative method’s iteration count depends on initial values and other factors.Source: §VI · p. 28
  • For three-hop AF MIMO capacity maximization, upper- and lower-bound robust designs have negligible performance gaps and perform much better than the non-robust design with channel errors.The simulations assume four antennas at all nodes and equal transmitter and receiver correlation factors across hops.Source: Figure 4 · p. 29, §VI · pp. 29–30
  • For three-hop MAX-MSE minimization, the two robust bound-based designs have the same performance and outperform the non-robust design when channel errors are large.MAX-MSE minimizes the largest diagonal element of the destination data-detection MSE matrix, rather than the sum, and can optimize the worst case across data streams.Source: Figure 5 · p. 30, §VI · p. 30

VII. CONCLUSIONS

  • The matrix-monotonic framework unifies linear and nonlinear transceiver and training designs, including robust counterparts, derives optimal diagonalizable matrix structures, simplifies optimization, and extends to multiple matrix variables.The conclusion identifies future applications including M2M multi-hop cooperative communications and distributed MIMO networks with mutual interference, such as dense heterogeneous networks.Source: §VII · p. 31

APPENDIX A PROOFS OF LEMMAS 3 AND 4 · APPENDIX B PROOFS OF MATRIX INEQUALITIES

Appendix A proves Lemmas 3 and 4 by bounding an objective and identifying conditions for attaining the bound. Appendix B proves matrix inequalities using optimization conditions, simultaneous diagonalization, and eigenvalue ordering.

APPENDIX A PROOFS OF LEMMAS 3 AND 4

  • The Appendix A construction defines ˜Π using the eigenvalues of Π and a scalar τ satisfying 0 < τ < [ΛΠ]NΠ,NΠ.Here NΠ = Rank(Π).Source: Appendix A · pp. 31–32
  • The upper bound holds with equality if and only if both equalities in (74) and (75) hold.Source: Appendix A · p. 32
  • The numerator reaches its maximum when U˜F = VΠ and Rank{˜F} ≤ NΠ; equality in (75) holds when U = UΠUH.The proof also states that VAin may be any unitary matrix because its value does not affect the objective function.Source: Appendix A · p. 33

APPENDIX B PROOFS OF MATRIX INEQUALITIES

  • The Appendix B proof formulates the left-hand inequality as an optimization problem and derives a KKT condition with a Hermitian Lagrange multiplier Φ.Source: Appendix B · p. 33
  • The proof uses QHQ = I and simultaneous diagonalization to reduce the minimum-value objective to one involving QHAQ and B.The passage states that QHAQ and B can be simultaneously diagonalized by a common unitary matrix.Source: Appendix B · pp. 33–34
  • The maximum and minimum reduce to choosing eigenvalue orders, which the proof determines using majorization and the Schur-convexity of 1/x.Source: Appendix B · p. 34

APPENDIX C THE DERIVATION OF OPTIMAL Q … 3) Case 6:

Appendix C derives optimal Q by treating complicated cases individually, with supplied passages showing inequality-based objective reformulations and conditions for attaining equality or a minimum.

APPENDIX C THE DERIVATION OF OPTIMAL Q

F HF) + I], (88)

  • For (88), the equality condition is based on Inequality 3.The supplied passage does not state the condition's equation.Source: F HF) + I], (88) · p. 34

2) Case 5:

  • In Case 5, the objective function is reformulated before applying an inequality to obtain a bound.The passage identifies this as the Case 5 derivation but does not include the reformulated expression.Source: 2) Case 5 · p. 35

F HF + I)|, (90)

  • Using the left inequality of (38), the passage directly derives a subsequent bound; the first equality is also based on |AB + I| = |BA + I|.Source: F HF + I)|, (90) · p. 35

F HF + I)]. (91)

  • The passage states that attaining the minimum value in (91) requires an equation to hold, based on Inequality 2, but gives no Case 5 designation.Source: F HF + I)]. (91) · p. 35

3) Case 6:

  • In Case 6, the left inequality of (37) is applied directly.The resulting expression is not included in the supplied passage.Source: 3) Case 6 · p. 35
  • Based on Inequality 1, equality in (93) holds when the following equation is satisfied.Source: 3) Case 6 · p. 35
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