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A Framework for Fractional Matrix Programming Problems with Applications in FBL MU-MIMO

T0 review · 2 major / 8 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read A single-loop surrogate framework solves fractional matrix programs that Dinkelbach-based solvers cannot handle.

desk verdict Worth engaging for the sum-of-fractional-functions solver and the FBL MU-MIMO applications, but the product-maximization claim rests on a false convexity lemma and needs a fix before publication. read the letter →

arxiv 2502.01077 v2 pith:MRWQC44T submitted 2025-02-03 eess.SP

Mohammad Soleymani , Eduard Jorswieck , Robert Schober , Lajos Hanzo This is my paper · ORCID
classification eess.SP MSC 90C2690C30
keywords fractionalmatrixprogrammingmajorization-minimizationfiniteblocklengthcodingMU-MIMOenergyefficiencylatencyminimizationreconfigurableintelligentsurfacespectral-energytradeoff
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes a general optimization framework for fractional matrix programming (FMP), where the objective or constraints are sums or products of arbitrary continuous non-negative fractional functions of complex matrix variables. The central claim is that both a generic minimization problem and a generic maximization problem can be solved by a single-loop iterative algorithm that converges to a stationary point, using majorization-minimization surrogates. This matters because standard solvers such as generalized Dinkelbach handle only single- or multiple-ratio fractional programs, while latency, energy efficiency, mean-square error, and spectral-energy efficiency tradeoff metrics in multi-user MIMO finite-block-length systems naturally involve sums or products of fractions. The paper demonstrates the framework on sum-delay, geometric-mean-delay, MSE, and SEE-tradeoff optimizations, including RIS-aided systems.

What carries the argument

The load-bearing machinery is the majorization-minimization (MM) surrogate construction combined with the inequality $x^2/y \geq 2\bar{x}x/\bar{y} - \bar{x}^2 y/\bar{y}^2$ (and its complex version), used to bound each fractional function from above or below. For minimization, each numerator $f$ is replaced by a convex upper bound and each denominator $g$ by a concave lower bound with matching value and first derivative at the current iterate; for maximization, each $f$ is minorized by a concave lower bound and each $g$ by a convex upper bound. Auxiliary variables $t$ and $u$ turn the fractional structure into quadratically constrained convex surrogates (4) and (9), which are solved once per iteration.

What would settle it

Run the minimization algorithm on a single-ratio fractional program with convex numerator, concave denominator, and a feasible set where the unique stationary point is known in closed form; if the iterates converge to a point whose projected gradient of the Lagrangian is nonzero, the stationarity claim fails. A simpler check is to construct any continuous pair $f,g$ satisfying the paper's surrogate conditions but where the limiting point of the iterates is not a KKT point of the original problem (1).

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Extended reading notes

Core claim

The paper's core discovery is that any fractional program whose objective and constraints are sums (or products) of continuous non-negative fractional functions $h_{mi}=f_{mi}/g_{mi}$ can be re-expanded into a sequence of convex surrogate problems by introducing auxiliary variables $t_{mi}, u_{mi}$ that bound $g_{mi}$ and $f_{mi}$ from below and above. Theorem 1 and Lemma 1 treat minimization: replace $f_{mi}$ by a convex upper bound (or directly $u^2$), $g_{mi}$ by a concave lower bound, and iterate the surrogate problem (4) to a stationary point of the original. Theorem 2 and Lemma 2 treat maximization by replacing each $f_{mi}/g_{mi}$ with the quadratic lower bound $2a_{mi}t_{mi}-a_{mi}^2 g_{mi}$ (with $a_{mi}=\sqrt{f_{mi}(X^{(z)})/g_{mi}(X^{(z)})}$) and minorizing non-concave $f_{mi}$ and non-convex $g_{mi}$ by matching concave lower and convex upper bounds. The resulting algorithms are single-loop, in contrast to twin-loop Dinkelbach implementations, and the surrogate functions satisfy the three MM conditions (equality of value and gradient at the current iterate, global majorization/minorization), which is what guarantees convergence to a stationary point.

Load-bearing premise

The user must be able to construct, for every numerator and denominator in the problem, a convex upper bound or concave lower bound that equals the original function and its first derivative at each iterate and majorizes or minorizes it everywhere on the feasible set.

Editorial extensions

If this is right

  • The framework solves sum-delay, geometric-mean-delay, sum/max MSE, weighted-sum-EE, geometric-mean-EE, and SEE-tradeoff problems in MU-MIMO with finite-block-length coding, none of which Dinkelbach-based algorithms can address directly.
  • Because the objective and constraints are arbitrary continuous functions of fractional functions, the same machinery extends to other network scenarios: hardware-impaired channels, imperfect CSI, rate-splitting, NOMA, and other RIS architectures, as the authors outline.
  • Single-loop implementation reduces implementation and iteration complexity compared to twin-loop Dinkelbach, with numerical comparisons showing similar objective values at convergence but smoother and faster convergence.
  • For the max-min ratio problems where the generalized Dinkelbach algorithm applies, the framework matches its stationary-point guarantee while removing the inner loop.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The surrogate construction is a parameter-free alternative to the quadratic transform, and it might also yield a unified view of Dinkelbach and Charnes-Cooper as special surrogate choices when the problem has a single ratio.
  • The method's practical power hinges on the user's ability to derive tight surrogates; one could build a library of certified surrogates for standard wireless metrics, turning the framework into a drop-in resource allocator.
  • A promising testable extension is to use the same bounds for stochastic or online versions, where the fractional metrics are estimated from samples, since the MM conditions only require local tightness.
  • The convergence claim is to a stationary point, not global optimality; on non-convex instances the quality of the limit point depends on initialization, an implicit limitation the paper does not quantify.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 8 minor

Summary. The paper develops a majorization-minimization (MM) and convex-concave procedure (CCP) framework for fractional matrix programs. It presents a generic minimization problem (1) and a generic maximization problem (6), where objective and constraints are sums of nonnegative fractional functions, and provides iterative surrogate problems (3)/(4) and (7)/(9) with auxiliary variables. It claims convergence to stationary points, single-loop implementation, and applicability to sums and products of fractional functions. The framework is instantiated for FBL MU-MIMO systems: sum/geometric-mean delay minimization, sum/maximum MSE minimization, channel-dispersion bounding, SEE tradeoff, weighted-sum EE, geometric-mean EE, weighted-sum SINR, and RIS-aided extensions. Numerical results compare with Dinkelbach-based algorithms.

Significance. The proposed single-loop treatment of sum-of-fractional-functions matrix programs, if correct, would be a useful addition to the fractional-programming toolbox for wireless resource allocation, and the FBL MU-MIMO applications are timely and well chosen. The paper clearly builds on prior published concave rate bounds [18], [57] rather than deriving the target results from themselves. However, the headline product-of-fractional-functions maximization claim rests on a false inequality (Lemma 10); this invalidates the geometric-mean EE application in Section IV-C as written. The sum-of-FFs results and the minimization-side product results appear plausible and are backed by standard tangent inequalities.

major comments (2)
  1. [Lemma 10 / Section IV-C, Eq. (92)] The inequality (92) is false. The function Q(x)=prod_k x_k^2 is not jointly convex on R_+^K; for K=2 its Hessian at (1,1) is [[2,4],[4,2]], which is indefinite, so the first-order lower bound used in the proof cannot hold. Concretely, with K=2, bar_x=(1,1), x=(0.5,3), the left-hand side of (92) is (0.5*3)^2=2.25, while the right-hand side is 1+2(0.5-1)+2(3-1)=4, violating the claimed inequality. This is not a presentation issue: the entire GMEE derivation in Section IV-C, specifically the replacement of (48) by the weighted sum in (49) with coefficients (50), rests on Lemma 10. Since the surrogate in (49) is not guaranteed to minorize the objective of (48), the monotone-ascent/stationary-point argument for the geometric-mean EE problem (47) is unsupported. The abstract's product-of-FFs maximization claim and the corresponding entries in Table II and Table VI therefore need either a valid proof or removal/re-scoping. The minimization-side product of delays in Section III-C (via Lemma 9) and all sum-of-FFs applications are not affected by this issue.
  2. [Section II, Theorems 1/2 and Lemmas 1/2] The claimed convergence to a stationary point is not established by the proofs as written. The appendices verify the standard MM tangent conditions for the surrogates, but they do not prove that the sequence generated by solving (3)/(4)/(7)/(9) has limit points, that these limit points are stationary points of the original constrained problem, or even that the objective and constraint functions are differentiable. The phrase 'arbitrary continuous' in (1) and (6) is insufficient for a stationary-point statement, which requires gradients and some regularity of the feasible set. Please state explicit assumptions (for example, differentiability on X, compactness of the feasible set or bounded sublevel sets, and a constraint qualification) and provide a complete convergence argument, or cite a theorem that covers the auxiliary-variable formulation used here.
minor comments (8)
  1. [Appendix B, Eq. (78a)] In the proof of Theorem 1, the reformulated problem (78a) is written as a maximization over {X}, t, u; since (1) is a minimization problem and u^2/t is an upper bound, the objective should be a minimization.
  2. [Section II-A, paragraph after Theorem 1] The sentence 'In this case, (7) is a convex OP' appears in the minimization subsection and should refer to (3), not the maximization surrogate (7).
  3. [Section III-C, Eq. (26)] The first factor in the definition of alpha_k should read (L_k/r_k({W^(z-1)}))^((1-K)/K); as written it uses L_i and r_k, which is dimensionally inconsistent.
  4. [Appendix F, Lemma 10 proof] The proof text says the bound is obtained for prod_k x_k^(1/K), but the lemma statement concerns prod_k x_k^2; the proof text is inconsistent even apart from the false inequality itself.
  5. [Section IV-C, Eqs. (46)-(47)] The geometric mean exponent is written as 1/k with the user index k; it should be 1/K, where K is the number of users.
  6. [Section II-B, Eqs. (9b) and (10b)] The right-hand sides of constraints (9b) and (10b) are missing; each constraint should be written with an explicit >= 0 (or the intended bound).
  7. [Algorithms I and II] The algorithms return {W^(*)} but the optimization variables are {X}; use {X^(*)} for consistency.
  8. [Introduction] There is a typo 'Dinkelabch' in the first introduction paragraph; it should be 'Dinkelbach'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the central MM/CCCP derivation is self-contained and does not reduce to its inputs.

full rationale

The paper's derivation chain is not circular. Theorems 1 and 2 (Section II, Appendices B and D) transform the fractional programs by introducing auxiliary variables and applying the first-order Taylor/CCCP inequality in Lemma 11 and the quadratic lower bound in (87); these are standard inequalities whose proofs are independent of the target stationary-point claims. The convergence argument for both generic algorithms rests on the MM conditions of Lemma 5, and the general-case Lemmas 1 and 2 explicitly require the user to supply convex/concave surrogate functions with matching values and first derivatives, which the paper acknowledges in Section II-C as a key step that can be challenging in many practical scenarios. That is a stated assumption, not a hidden circular premise. The application-specific bounds reused from prior work, notably Lemma 3 and Lemma 8 from [18] and Lemma 11 from [57], are previously published mathematical inequalities with stated assumptions; although the author sets overlap with the present paper, these results are externally checkable and do not assume the framework's convergence or the inequalities being proved. No fitted parameter is relabeled as a prediction, and no uniqueness theorem is imported from the authors' own prior work to force a choice. The serious concern raised about Lemma 10 - that the product function Q_k x_k^2 is not jointly convex, so inequality (92) can fail - is a mathematical validity issue affecting the geometric-mean-EE surrogate, not a circularity issue: even if the lemma is false, the framework's derivation is not equivalent by construction to its own inputs. Therefore the appropriate circularity finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The framework itself is self-contained given standard MMA/CCP results, but its practical instantiations import domain assumptions about FBL coding, channel models, and the availability of hand-crafted surrogates. No new physical entities are introduced and no numerical constants are fitted to data.

assumptions (5)
  • standard math Iteratively solving a surrogate that is tight, tangent, and dominating at each iterate yields a stationary point of the original nonconvex problem (Lemma 5, Appendix A).
    This is the standard majorization-minimization convergence argument, adopted from [58].
  • domain assumption The functions f_mi and g_mi are continuous, with f_mi >= 0 and g_mi > 0, and are sufficiently differentiable for first-order surrogates to apply.
    The generic problem statement in Section II assumes arbitrary continuous functions, but stationary-point and Taylor arguments implicitly require differentiability and are not stated as an explicit hypothesis.
  • ad hoc to paper For each fractional function, a global convex or concave surrogate with matching value and first derivative exists and is available to the algorithm designer.
    Lemmas 1 and 2 depend on this availability, and the paper acknowledges in Section II-C that constructing such surrogates is a key and challenging step.
  • domain assumption The finite block length normal approximation for the rate (14) and the channel dispersion (16) are accurate models for the applications.
    The rate and dispersion expressions are imported from [18] and [21] and are not re-derived or validated in this paper.
  • domain assumption The wireless system model assumes Gaussian signaling, perfect CSI, treating interference as noise, and a convex power constraint set X.
    These assumptions are stated in Sections III, IV, and V and are standard in the cited prior work.

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Pith. "Pith review of A Framework for Fractional Matrix Programming Problems with Applications in FBL MU-MIMO." pith.science (2026). https://pith.science/paper/MRWQC44T

@misc{pith2026250201077,
  author       = {Pith},
  title        = {Pith review of: A Framework for Fractional Matrix Programming Problems with Applications in FBL MU-MIMO},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MRWQC44T}},
  note         = {Machine review of arXiv:2502.01077}
}
read the original abstract

An efficient framework is conceived for fractional matrix programming (FMP) optimization problems (OPs) namely for minimization and maximization. In each generic OP, either the objective or the constraints are functions of multiple arbitrary continuous-domain fractional functions (FFs). This ensures the framework's versatility, enabling it to solve a broader range of OPs than classical FMP solvers, like Dinkelbach-based algorithms. Specifically, the generalized Dinkelbach algorithm can only solve multiple-ratio FMP problems. By contrast, our framework solves OPs associated with a sum or product of multiple FFs as the objective or constraint functions. Additionally, our framework provides a single-loop solution, while most FMP solvers require twin-loop algorithms. Many popular performance metrics of wireless communications are FFs. For instance, latency has a fractional structure, and minimizing the sum delay leads to an FMP problem. Moreover, the mean square error (MSE) and energy efficiency (EE) metrics have fractional structures. Thus, optimizing EE-related metrics such as the sum or geometric mean of EEs and enhancing the metrics related to spectral-versus-energy-efficiency tradeoff yield FMP problems. Furthermore, both the signal-to-interference-plus-noise ratio and the channel dispersion are FFs. In this paper, we also develop resource allocation schemes for multi-user multiple-input multiple-output (MU-MIMO) systems, using finite block length (FBL) coding, demonstrating attractive practical applications of FMP by optimizing the aforementioned metrics.

Figures

Figures reproduced from arXiv: 2502.01077 by the authors.

Figure 1
Figure 1. Examples of iterative MM-based algorithms. • ∂f˜mi({X}) ∂{X} [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. SEE tradeoff versus P for NBS = Nu = 4, and K = 3. A. Sum Delay Minimization The optimization of {W} for fixed Θ is provided in Section III-B, when the sum delay is minimized. Indeed, solving (22) gives {W(z)}. Here, we solve the sum delay minimization for fixed {W}, which can be written as min Θ∈D X k Lk rk . (56) To solve (56), we employ an approach similar to the one in Section III-B. More specifically, we levera… view at source ↗
Figure 3
Figure 3. Average sum EE obtained by solving (44) and (60) versus P for NBS = Nu = 5, and K = 2. {W(z)} is calculated by solving (45). Upon employing the concave lower bound rˆk, we can compute Θ(z) by solving max Θ∈D X k αkrˆk Ps + ηTr(W(z) k W(z)H k ) , (60) converging to an SP of the sum EE maximization problem. In [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Average sum delay, using the metric in (19), versus P for NBS = 4, Nu = 3, and K = 2. θmn = 0, ∀m ̸= n} , (70) where the ϕis for all i are the only possible phase shifts that can be tuned [27]. To obtain a suboptimal solution for D3, we first relax the assumption of ha…
Figure 5
Figure 5. Figure 5: Comparison of our framework and Dinkelbach-based algo￾rithms for P = 10 dB, NBS = Nu = 5, and K = 2. these examples, our framework performs very similarly to the Dinkelbach-based algorithms. However, the Dinkelbach-based algorithms cannot solve any of the OPs considere…
Figure 7
Figure 7. Figure 7: Simulation setup. Lemma 7 ([60]). Consider arbitrary matrices Γ ∈ C m×n and Γ¯ ∈ C m×n, and positive definite matrices Ω ∈ C m×m and Ω¯ ∈ C m×m, where m and n are arbitrary natural numbers. Then, we have: ln [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.