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Surrogate-Enhanced Fractional Programming for MIMO Device-to-Device Interference Networks

Zihan Jiao, Xinping Yi, Shi Jin, Giuseppe Caire

arXiv:2609.36586v2cs.ITeess.SPmath.IT

TL;DR

Extending scalar SEFP to weighted sum-log-determinant optimization with matrix-valued ratios is nontrivial because the ratio and auxiliary matrix need not commute. The paper lifts scalar RIT along Hermitian auxiliary-matrix eigendirections, combines it with matrix QT, and uses the resulting matrix SEFP framework to develop SEFPLinQ. In the 8 × 8 case, at iteration 100, SEFPLinQ’s average sum-rate increment is 13,068.2 Mbps versus 11,128.7 Mbps for FPLinQ, a 17.43% improvement.

  • Problem

    Extending scalar SEFP to weighted sum-log-determinant optimization with matrix-valued ratios is nontrivial because the ratio and auxiliary matrix need not commute.

    Source: §I · p. 2, B. Why is the Scalar-to-Matrix… · p. 3

  • Method

    The paper lifts scalar RIT along Hermitian auxiliary-matrix eigendirections, combines it with matrix QT, and uses the resulting matrix SEFP framework to develop SEFPLinQ.

    Source: §I · p. 2

  • Results

    In the 8 × 8 case, at iteration 100, SEFPLinQ’s average sum-rate increment is 13,068.2 Mbps versus 11,128.7 Mbps for FPLinQ, a 17.43% improvement.

    Source: C. Flexible-association D2D Networks · pp. 12–13

  • Takeaways & Limitations

    The paper reports state-of-the-art weighted sum-rate performance for SEFPLinQ in flexible-association MIMO D2D networks.

    Source: §VII · p. 13

Abstract

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Interference management in multi-stream multi-input multi-output (MIMO) device-to-device (D2D) networks often leads to weighted sum-rate maximization with sum-log-determinant involving matrix-valued signal-to-interference-plus-noise ratios (SINRs). The state-of-the-art paradigms, including weighted minimum mean-square error (WMMSE) and fractional programming (FP), have achieved tremendous success in link scheduling, power control, and beamforming problems. Recently, an upgraded FP approach, named surrogate-enhanced FP (SEFP) in a scalar form, has attained improved performance in joint uplink scheduling and power control in coordinated multicell SISO networks. The proposed SEFP improves the surrogate construction of the classical Lagrangian dual transform plus quadratic transform for logarithmic fractional objectives with a novel reciprocal-inverse transform (RIT), yet its extension to the matrix form for MIMO settings does not seem straightforward because the matrix ratio and the auxiliary matrix generally do not commute. In this paper, by leveraging mathematical tools from Hermitian functional calculus, we extend RIT to matrix ratios by lifting the scalar RIT along the eigendirections of an auxiliary matrix and recasting the resulting directional construction in an operator form. Following, we develop a matrix SEFP framework for weighted sum-log-determinant maximization. Further, we establish a unified view of SEFP and the recently proposed XMMSE method, specify when the two algorithms attribute to identical minorization-maximization (MM) surrogates and variable update trajectories, and prove XMMSE can be considered as a special case of SEFP under the algorithmic family perspective. Inspired by the unified view, we develop SEFPLinQ for the joint scheduling and beamforming optimization in flexible-association MIMO D2D networks. Numerical results demonstrate consistent performance gains.

I. INTRODUCTION

The paper addresses weighted sum-rate maximization in MIMO D2D networks, where coupled interference and matrix-valued SINRs make optimization difficult. It introduces scalar SEFP and motivates its nontrivial matrix extension, alongside a framework for relating SEFP to XMMSE.

I. INTRODUCTION

  • MIMO D2D weighted sum-rate maximization is notoriously difficult: interference couples each link with many others, creating highly nonconvex, generally NP-hard optimization, while multi-antenna multi-stream rates use log-determinants of matrix-form SINRs,.Source: §I · p. 1
  • The paper asks whether SEFP’s tighter scalar surrogate can extend to weighted sum-log-determinant objectives with matrix-valued ratios, and how SEFP relates to XMMSE.Scalar SEFP replaces the classical LDT with RIT; the matrix extension is nontrivial because the ratio and auxiliary matrix need not commute.Source: §I · p. 2
  • Matrix SEFP lifts scalar RIT along eigendirections of a Hermitian auxiliary matrix and combines it with matrix QT for weighted sum-log-determinant maximization.The construction uses Hermitian functional calculus and expresses the directional lift in operator form.Source: §I · p. 2
  • SEFP and XMMSE have losslessly interconvertible auxiliary variables and identical MM surrogates under exact updates, while SEFP retains additional ratio-decoupling and update-order flexibility.This unified view motivates developing SEFPLinQ for joint scheduling and multi-stream beamforming in flexible-association MIMO D2D networks.Source: §I · p. 2

II. PRELIMINARIES

  • Classical FP has been applied to MIMO D2D joint scheduling and multi-stream beamforming, while scalar SEFP improved joint uplink scheduling and power control in SISO D2D networks.These prior applications establish the settings that motivate the proposed MIMO SEFP extension.Source: §II · pp. 2–3

A. The Scalar RIT and SEFP [17]

  • For scalar objectives, RIT gives a lower-bound surrogate for log(1+r) for r≥0, with equality exactly when α=r for r>0; QT introduces auxiliary variables.Source: A. The Scalar RIT and SEFP [17] · p. 3

B. Why is the Scalar-to-Matrix Extension Nontrivial?

A direct scalar-to-matrix RIT substitution fails because the matrix ratio and auxiliary matrix need not commute. Hermitian functional calculus provides a way to lift scalar lower bounds along the spectral directions of a Hermitian matrix.

B. Why is the Scalar-to-Matrix Extension Nontrivial?

  • Replacing the scalar auxiliary variable with a matrix does not preserve the scalar derivation: the ratio and auxiliary matrix need not commute or be simultaneously diagonalizable.The scalar-to-matrix extension therefore cannot mechanically follow the route used for FP.Source: B. Why is the Scalar-to-Matrix… · p. 3
  • Because log det(Id + R) sums log(1+λi(R)) over spectral directions, a matrix RIT must allow different scalar contact values in different directions.The auxiliary matrix is chosen Hermitian positive definite so its eigenvalues provide contact values and its eigenvectors specify the directions of the lift.Source: B. Why is the Scalar-to-Matrix… · p. 3

C. Hermitian Functional Calculus

  • Hermitian functional calculus maps scalar functions on a Hermitian matrix’s spectrum to matrix functions, preserving pointwise inequalities as matrix inequalities.Functions of the same Hermitian matrix commute, while functions of different matrices need not; parameter integration can also be performed eigenvalue-wise.Source: C. Hermitian Functional Calculus · p. 4
  • Hermitian functional calculus can turn nonnegative or positive scalar functions on the spectrum into positive semidefinite or positive definite matrix functions, supporting a matrix lower bound for the logarithm.The paper uses this property in its matrix RIT construction.Source: C. Hermitian Functional Calculus · p. 4

III. MATRIX RECIPROCAL-INVERSION TRANSFORM AND SURROGATE-ENHANCED FRACTIONAL PROGRAMMING

The paper constructs a matrix RIT and combines it with matrix QT to obtain matrix SEFP for weighted sum-log-determinant optimization. The resulting auxiliary-variable formulation supports an MM interpretation.

A. The Matrix RIT

  • Matrix RIT extends scalar RIT to weighted sum-log-determinant objectives with matrix ratios using Hermitian functional calculus.Its operator-form representation is introduced to facilitate combination with matrix QT.Source: A. The Matrix RIT · p. 4
  • For a positively weighted term with positive-definite ratio Rm, the matrix RIT has a unique optimal auxiliary matrix; when Rm is singular, a family of maximizing matrices is available.The singular case permits any matrix of the stated form with positive ηm,j values.Source: A. The Matrix RIT · p. 4

B. Matrix SEFP = Matrix RIT + QT

  • Successively applying matrix RIT and matrix QT yields an equivalent matrix SEFP formulation for the weighted sum-log-determinant problem.The QT auxiliary-variable update is unique for fixed x and Γm.Source: B. Matrix SEFP = Matrix RIT + QT · p. 5

C. MM Interpretation

The paper frames matrix SEFP and XMMSE within an MM/BCA perspective and establishes their surrogate-level connection. XMMSE’s auxiliary-variable representation also provides a functional path that can be factored through SEFP variables.

C. MM Interpretation

  • Fixing the auxiliary variables at their exact maximizers makes the BCA objective a globally valid minorizing surrogate, tight at the current iterate.This is the MM interpretation used for matrix SEFP.Source: C. MM Interpretation · p. 5

IV. INTERCONNECTIONS WITH XMMSE

  • The section completes the connection between SEFP and XMMSE, paralleling the established relationship between FP and WMMSE.It places the four algorithms in a common interconnection picture.Source: §IV · p. 5, Figure 1 · p. 5
  • The XMMSE auxiliary-variable path can be losslessly factored through SEFP variables, and the two representations induce the same MM surrogate on the optimal auxiliary-variable manifold.The connection is established through a functional auxiliary-variable formulation and a finite-dimensional factorization.Source: §IV · p. 5

A. Functional Auxiliary-Variable Representation of XMMSE

  • XMMSE is recast as a functional auxiliary-variable representation of the generic weighted sum-log-determinant problem without invoking its MSE interpretation.The transform-induced representation recovers the original objective when its auxiliary variables are maximized.Source: A. Functional Auxiliary-Variable… · p. 6

B. Two Coordinates Mapping Under the Optimal Auxiliary Variable Manifold

At optimal auxiliary variables, matrix SEFP and XMMSE are interchangeable coordinate representations and induce the same MM surrogate. Their original-variable updates coincide under exact updates and shared solution-selection conditions, while SEFP retains broader algorithmic flexibility.

B. Two Coordinates Mapping Under the Optimal Auxiliary Variable Manifold

  • SEFP and XMMSE auxiliary variables are interchangeable coordinate representations on their optimal manifold, including when the SINR matrix is rank deficient.The SEFP-induced variables provide a lossless finite-dimensional factorization of the XMMSE auxiliary-variable path.Source: B. Two Coordinates Mapping Under the… · p. 7
  • With exact auxiliary-variable updates, matrix SEFP and XMMSE construct identical MM surrogates and original-variable updates under the same initialization and maximizer selection.The update equivalence assumes exact surrogate-subproblem solutions and matching choices when a subproblem has multiple maximizers.Source: B. Two Coordinates Mapping Under the… · p. 7
  • XMMSE replaces WMMSE’s explicit MSE weight matrices with an exact integral rate–MSE identity and an equivalent sum-regularized-MSE reformulation.SEFP instead revisits the Lagrangian dual transform bound through reciprocal-inversion coordinates, separating a structural coupling in that bound.Source: B. Two Coordinates Mapping Under the… · p. 7
  • The equivalence applies to utility maximization with fixed utility terms and weights, but scheduling-dependent terms and weights leave SEFP additional flexibility.Scheduling can change both the active utility term and its associated weight.Source: B. Two Coordinates Mapping Under the… · p. 7

C. XMMSE as a Special SEFP Instance: From Continuous to Mixed-Discrete Optimization

XMMSE is one member of the broader SEFP algorithm family, arising from a particular ratio decoupling and block-coordinate update order. SEFP’s additional choices matter for mixed-discrete scheduling because they can expose an edge-separable objective.

C. XMMSE as a Special SEFP Instance: From Continuous to Mixed-Discrete Optimization

  • XMMSE is a particular SEFP instance under a specific ratio decoupling and block-coordinate update order.SEFP inherits nonunique ratio decouplings and block-selection rules from FP.Source: C. XMMSE as a Special SEFP Instance:… · p. 7
  • Different admissible block-coordinate orders generate distinct SEFP variants, while one mixed-level rule reproduces XMMSE.Alternative coordinate realizations of the quadratic-transform block maximize the same RIT-induced quantity for fixed Γm.Source: C. XMMSE as a Special SEFP Instance:… · p. 8
  • When scheduling makes utility terms and weights decision-dependent, an appropriate SEFP ratio decoupling exposes edge-wise separability, unlike the described XMMSE representation.In XMMSE, candidate identity and network interference couple the continuum of auxiliary variables associated with each candidate link.Source: C. XMMSE as a Special SEFP Instance:… · p. 8

V. JOINT SCHEDULING, AND BEAMFORMING VIA MATRIX SEFP

The paper applies matrix SEFP to joint link scheduling and beamforming in flexible-association MIMO interference networks. Its scaling reformulation preserves the original optimization while extending continuously to rank-deficient SINR matrices.

V. JOINT SCHEDULING, AND BEAMFORMING VIA MATRIX SEFP

  • The framework applies matrix SEFP to the joint link-scheduling and beamforming weighted-sum-rate problem in a flexible-association MIMO interference network.The comparison is set up against FPLinQ under the setup described in [11, Section V].Source: §V · p. 8, A. System Model and Problem Formulation · p. 9

A. System Model and Problem Formulation

  • The joint scheduling and beamforming problem is a mixed discrete–continuous nonconvex optimization that is NP-hard.The flexible-association network schedules a transmitter for each receiver or leaves that receiver inactive.Source: A. System Model and Problem Formulation · pp. 8–9

1) Matrix SEFP reformulation and the scaling version:

  • The scaled SEFP formulation is an exact reparameterization that preserves the objective and scheduling and beamforming variables.Its spectral functions are continuous at zero eigenvalues, so the objective extends to rank-deficient SINR matrices; the fixed-variable scaled auxiliary subproblem remains strictly concave.Source: 1) Matrix SEFP reformulation and the… · p. 9

2) Update of the auxiliary variables:

The auxiliary updates use the current SINR matrix and a first-order condition for the scaled variable. The resulting edge-wise surrogate has a receiver-specific quadratic coefficient, unlike the cited FPLinQ coefficient.

2) Update of the auxiliary variables:

  • For fixed scheduling and beamformers, the RIT-induced auxiliary variable is updated using the current SINR matrix.For fixed auxiliary Γ, the first-order condition for the scaled variable Zj yields its update.Source: 2) Update of the auxiliary variables · pp. 9–10
  • The first-order condition for Zj is a generalized Sylvester equation with a closed-form solution under the RIT-induced auxiliary update.The stated identity and matrix-function property establish that solution.Source: 2) Update of the auxiliary variables · p. 10

3) Edge-wise decomposition of the SEFP surrogate:

  • Unlike FPLinQ’s transmitter-indexed quadratic coefficient, SEFPLinQ’s Aji generally depends on both transmitter and candidate receiver.The distinction follows from receiver-specific δ(Γj) weighting for intended signals versus identity weighting when signals act as interference.Source: 3) Edge-wise decomposition of the SEFP… · p. 10

4) Tentative beamformer for each candidate edge and the first step matching:

The first matching uses edge-specific tentative beamformers and scores to separate candidate-edge decisions before enforcing network-wide matching constraints. A second matching then adjusts transmitter–receiver service relations against the original weighted sum rate.

4) Tentative beamformer for each candidate edge and the first step matching:

  • Candidate-edge tentative beamformers follow the transmit-power constraint and a first-order KKT condition, with the multiplier obtained by one-dimensional bisection.Source: 4) Tentative beamformer for each… · p. 10
  • The first matching uses edge weights derived from tentative beamformers and solves a maximum weighted bipartite matching problem, which admits polynomial-time optimal algorithms [22] and.Source: 4) Tentative beamformer for each… · p. 10
  • With Γ and Z fixed, each receiver reuses its auxiliary pair across candidate transmitters, yielding independently computed tentative beamformers and scalar scores before matching.The remaining coupling is imposed by the one-to-one matching constraints.Source: 4) Tentative beamformer for each… · p. 10

5) The second step matching:

4) Tentative beamformer for each candidate edge and the first step matching:

  • Recomputing tentative beamformers for every feasible edge independently of current beamformers and schedules can reactivate a transmitter after its beamformer becomes zero, when a candidate coefficient is nonzero.Source: 4) Tentative beamformer for each… · p. 11

5) The second step matching:

  • The second matching updates transmitter–receiver service relations by directly optimizing the original weighted sum-rate objective over a weighted bipartite matching.Source: 5) The second step matching · p. 11

VI. SIMULATION RESULTS

Across fixed and flexible-association D2D simulations, SEFPLinQ generally improves on FPLinQ, with stronger gains in larger MIMO configurations and faster convergence. The simulations also compare SEFP with XMMSE and report matching trajectories under fixed single-association settings; the conclusion places SEFPLinQ within the matrix SEFP framework.

A. Common Simulation Setup

  • The simulations use common D2D parameters, compare fixed and flexible association, and evaluate algorithms on the same network and channel realizations across 30 independent channel realizations.The flexible setting has 100 receivers, 300 transmitters, and 400 feasible candidate edges; each transmitter and receiver can participate in at most one active link per scheduling interval.Source: A. Common Simulation Setup · p. 11

B. Fixed Single-association D2D Networks

  • In fixed single-association SISO networks, SEFPLinQ consistently achieves the highest sum rate across tested network sizes, slightly but steadily outperforming FPLinQ and substantially outperforming heuristic scheduling baselines.For both SISO and multi-data-stream MIMO, SEFP and XMMSE trajectories coincide within numerical precision, as do FP and WMMSE; SEFP and XMMSE visibly outperform closed-form FP and WMMSE.Source: B. Fixed Single-association D2D Networks · pp. 11–12
  • Under PF weight updating with 100 fixed-association links, SEFPLinQ shifts the rate CDF toward higher low- and medium-range rates while maintaining competitive high-rate performance, yielding better link balance and higher overall log-utility.The passage attributes this fairness-setting benefit to the SEFP surrogate.Source: B. Fixed Single-association D2D Networks · p. 12

C. Flexible-association D2D Networks

  • In flexible-association PF scheduling, SEFPLinQ’s rate CDF is generally better than FPLinQ’s in SISO and 2 × 2 MIMO, while both avoid BCD’s performance loss from premature turning-off.The second matching is indispensable for SEFPLinQ and FPLinQ in this setting.Source: C. Flexible-association D2D Networks · p. 12

VII. CONCLUSION

  • The conclusion presents matrix SEFP as a framework for weighted sum-log-determinant maximization and identifies XMMSE as a particular realization within its algorithmic family under exact auxiliary-variable updates.It reports that SEFPLinQ, built on matrix SEFP, achieves state-of-the-art weighted sum-rate performance in flexible-association MIMO D2D networks.Source: §VII · p. 13

APPENDIX A PROOF OF THEOREM 1

Appendix A constructs matrix RIT by lifting scalar RIT bounds along the eigendirections of an auxiliary matrix and assembling them into an operator form. It establishes a global log-determinant lower bound and proves the auxiliary operator used in the construction is invertible.

APPENDIX A PROOF OF THEOREM 1

  • The scalar RIT lower bound is lifted along Γ’s eigendirections, with eigenvalues setting contact values and eigenvectors specifying the directions.The construction aims to assemble these directional parameters into the matrix Γ without retaining the individual directions in the final transform.Source: Appendix A · pp. 13–14
  • For every fixed Γ ≻ 0, ΦΓ(R) is a globally valid lower bound of the log-determinant.For R ≻ 0, equality holds if and only if Γ = R; if R is singular, every Γ of the form (20) attains equality.Source: Appendix A · p. 14
  • A common matrix WΓ recovers each direction-wise scalar RIT term through its operation along the corresponding eigenvector, yielding the operator representation used in Theorem 1.Summing the directional identities and applying the stated lemma gives the transform; applying it termwise and weighting by ωm yields (17).Source: Appendix A · p. 14

APPENDIX B PROOF OF COROLLARY 2

  • DΓ is strictly positive definite and therefore invertible, supporting the unique matrix WΓ used in the construction.The appendix establishes the corresponding corollary by summing the identities over m with weights ωm.Source: Appendix B · p. 14, Appendix A · p. 14

APPENDIX C PROOF OF LEMMA 3

Appendix C proves the matrix RIT lower-bound property, including its equality conditions. Appendices D and E establish the stated SEFP equivalence and show that the XMMSE-induced auxiliary variable lies in the structured family without loss at the optimum.

APPENDIX C PROOF OF LEMMA 3

  • For R ≻ 0, equality in the matrix RIT lower bound requires Γ = R, while for singular R, every Γ of the form (20) attains equality.The proof lifts the scalar RIT inequality eigenvalue-wise, then uses the equality condition to establish uniqueness for positive-definite R.Source: Appendix C · pp. 14–15

APPENDIX D PROOF OF THEOREM 2

  • Applying Lemma 1 to the auxiliary-variable expression yields the objective in (28), and combining it with Theorem 1 proves Theorem 2’s equivalence.The resulting update (29) follows directly from (25); the auxiliary matrix is also rewritten as the unique solution of a generalized Sylvester equation.Source: Appendix D · p. 15
  • Substituting (18) for nonsingular Rm and (20) for singular Rm gives (30) from the generalized Sylvester equation.The appendix explicitly distinguishes the substitutions by whether Rm is singular.Source: Appendix D · p. 15

APPENDIX E PROOF OF THEOREM 3

  • Restricting U to the structured family (41) causes no loss at the optimum because it contains the unique pointwise XMMSE-induced auxiliary variable.Together with (42) and the pointwise SEFP equivalence in (31), this proves (44).Source: Appendix E · p. 15
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