{"id":"e32ce607-bd61-4409-8ea6-5eae26c03405","arxiv_id":"2502.01077","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A single-loop majorization-minimization framework solves fractional matrix programs with sums or products of ratios, demonstrated on delay, energy-efficiency, MSE, and SEE-tradeoff problems in finite-block-length MU-MIMO systems.","lead":"This paper proposes a general optimization framework for problems whose objective or constraints contain sums or products of ratios of matrix functions, then applies it to resource allocation in multi-user MIMO systems using short-packet (finite block length) coding.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 10, used for geometric-mean-of-EE maximization, is false: the claimed lower bound (92) fails, so the surrogate (49) need not minorize the product objective and the stationary-point guarantee for product-of-fractional-functions maximization is unsupported.","rationale":"I read the paper's central claim as the ability to converge, via single-loop MM-type surrogates, to stationary points of fractional matrix programs with sums and products of arbitrary continuous non-negative fractional functions. The sum-side machinery in Theorems 1-2 and Lemmas 1-2 follows the standard CCP/quadratic-transform template and appears internally sound: the auxiliary-variable transformations preserve feasibility, the convexified surrogates are global bounds with matching values and gradients at the current point, and the explicit sum/delay and sum-MSE examples are coherent. The reader's identified weakest assumption, that suitable convex/concave surrogates for f and g must exist, is a genuine practical limitation but is acknowledged by the paper and is a matter of applicability, not correctness. The more severe, and more concrete, problem is that the maximization of a product of fractional functions, showcased as a distinguishing capability, is proved through Lemma 10, and Lemma 10 is false. The counterexample is elementary: the function x1^2 x2^2 is not convex on the positive orthant, so its first-order Taylor expansion is not a lower bound. Since Section IV-C transforms the geometric-mean objective into Q_k e_k^2 and then uses exactly that invalid first-order lower bound, the resulting algorithm has no proven monotonicity or stationary-point guarantee for the actual GMEE objective. This does not invalidate the central sum-of-FF framework, which is the largest portion of the paper, but it does invalidate one of the explicit claims of novelty over Dinkelbach-based solvers. A conditional acceptance is appropriate: the authors should either correct Lemma 10 and the GMEE proof or remove the product-maximization claim from the paper's advertised scope.","tokens_in":29133,"tokens_out":14168,"duration_ms":150559,"concrete_test":"Analytically verify Lemma 10 inequality (92) at K=2 with bar_x_1=bar_x_2=1, x_1=0.5, x_2=3; the inequality fails since LHS=2.25 < RHS=4. Then, as a settlement test for Section IV-C, run the proposed GMEE-maximization algorithm on a simple two-user power-allocation instance and record the true objective Q_k (r_k/p_k)^{1/K} at each iteration; if it is not monotonically non-decreasing, the surrogate (49) is not a valid MM minorizer.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's headline claim includes solving problems whose objectives or constraints are products of fractional functions, with geometric-mean EE maximization in Section IV-C as a concrete instance. The derivation of the GMEE algorithm rests on Lemma 10, which asserts that Q_k x_k^2 is jointly convex for x_k>=0 and therefore that inequality (92), Q_k x_k^2 >= Q_k bar_x_k^2 + sum_k alpha_k (x_k-bar_x_k), holds with alpha_k = 2 bar_x_k prod_{i!=k} bar_x_i^2. This assertion is false. For K=2, bar_x=(1,1), and x=(0.5,3), the left-hand side is (0.5*3)^2 = 2.25, while the right-hand side is 1 + 2(0.5-1) + 2(3-1) = 4, so (92) is violated. Equivalently, the Hessian of x1^2 x2^2 is indefinite, so the function is not jointly convex. Because (92) is the only mechanism that turns the product objective into the weighted sum of fractional functions used in (49), the surrogate objective in (49) is not guaranteed to be a global lower bound on the GMEE objective. Consequently, the monotone-ascent argument for convergence to a stationary point of (47) breaks down. This is distinct from the acknowledged challenge of finding surrogate functions in Section II-C: the issue here is an internally invalid inequality, not a missing user-supplied surrogate. The minimization-side product example in Section III-C, which uses the valid concavity of the geometric mean, is not affected, nor are the sum-of-fractional-functions applications. But the maximization product claim, advertised in the abstract and in Table I/II, lacks a correct proof unless Lemma 10 is replaced.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":29541,"tokens_out":12584,"duration_ms":130817,"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":[{"comment":"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.","section":"Lemma 10 / Section IV-C, Eq. (92)"},{"comment":"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.","section":"Section II, Theorems 1/2 and Lemmas 1/2"}],"minor_comments":[{"comment":"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.","section":"Appendix B, Eq. (78a)"},{"comment":"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).","section":"Section II-A, paragraph after Theorem 1"},{"comment":"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.","section":"Section III-C, Eq. (26)"},{"comment":"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.","section":"Appendix F, Lemma 10 proof"},{"comment":"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.","section":"Section IV-C, Eqs. (46)-(47)"},{"comment":"The right-hand sides of constraints (9b) and (10b) are missing; each constraint should be written with an explicit >= 0 (or the intended bound).","section":"Section II-B, Eqs. (9b) and (10b)"},{"comment":"The algorithms return {W^(*)} but the optimization variables are {X}; use {X^(*)} for consistency.","section":"Algorithms I and II"},{"comment":"There is a typo 'Dinkelabch' in the first introduction paragraph; it should be 'Dinkelbach'.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":"The central concern is Lemma 10: the product-of-FFs maximization claim, which appears in the abstract, in Section II-C, and in the GMEE application of Section IV-C, is based on a demonstrably false inequality. The rest of the framework, especially the sum-of-FFs minimization and maximization solvers, is plausible and useful. If the authors cannot supply a valid minorization of the product objective, they should remove the product-maximization claims and revise the abstract and contribution statements accordingly. The paper may also need a more rigorous convergence statement for the generic theorems."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper delivers a genuinely useful single-loop framework for sums of fractional matrix functions, and the applications to FBL MU-MIMO are concrete and well executed. But the advertised product-of-fractional-functions maximization rests on a false lemma, so the geometric-mean-EE algorithm and the related claims are not currently supported.\n\nThe minimization side (Theorem 1/Lemma 1) uses standard MM/CCP surrogates and looks defensible. The sum-maximization side (Theorem 2/Lemma 2) is a clean joint-convex alternative to Shen-Yu's alternating quadratic transform, and the single-loop feature is a real practical improvement. The numerical comparison with Dinkelbach is honest: similar performance, fewer iterations, no dramatic overclaiming.\n\nNow the soft spot. Lemma 10 (Appendix F) asserts that product_k x_k^2 is jointly convex for x_k >= 0, giving the linear lower bound (92). That is false for K >= 2. The Hessian of x1^2 x2^2 is indefinite, and the bound fails even at simple points: with bar_x=(1,1) and x=(0.5,3), LHS is 2.25 while RHS is 4. The proof of Lemma 10 is also garbled—it says it obtains a lower bound for product x_k^{1/K}, which is a different function. Since (49) in Section IV-C relies on (92) to minorize the geometric-mean-EE objective, the monotone-ascent argument for (47) collapses. This is not the \"finding surrogate functions is hard\" caveat; it is an internally invalid inequality.\n\nWhat survives: the minimization of sum/geometric-mean of delays (Section III-B/C) uses the valid concavity of the geometric mean via Lemma 9, so that part is fine. The sum-of-FF applications (SEE tradeoff, weighted sum EE, sum SINR) are healthy. The product-maximization claim, emphasized in the abstract and Table I/II, needs a corrected lemma or a restriction to cases where a valid minorization exists.\n\nMinor stuff: several typos and cross-reference errors, and the Monte Carlo curves lack error bars, but that is secondary.\n\nBottom line: this is a serious paper in its sum-FF and minimization parts, and the flaw is local and fixable. The authors should replace or repair Lemma 10, or narrow the claims. It deserves peer review—a competent referee will catch this, and the paper is worth engaging with once corrected.","headline":"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.","tokens_in":30030,"tokens_out":3678,"would_cite":true,"duration_ms":34850,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C26","90C30"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single-loop surrogate framework solves fractional matrix programs that Dinkelbach-based solvers cannot handle.","keywords":["fractional matrix programming","majorization-minimization","finite block length coding","MU-MIMO","energy efficiency","latency minimization","reconfigurable intelligent surface","spectral-energy efficiency tradeoff"],"falsifier":"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).","tokens_in":28953,"feed_emoji":"📡","tokens_out":4554,"duration_ms":41272,"temperature":0.7,"pith_summary":"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.","feed_headline":"Single-loop solver tackles fractional matrix programs Dinkelbach can't","feed_subtitle":"Covers sums and products of latency, energy efficiency, and MSE metrics in MU-MIMO with finite-block-length coding.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Shen and Yu's quadratic transform is the main baseline for maximization FMP; the paper contrasts its alternating-optimization structure with its own single-loop surrogate approach.","marker":"[10]"},{"why":"Crouzeix and Ferland's generalized Dinkelbach algorithm is the twin-loop baseline for multiple-ratio max-min FMP that the framework compares against.","marker":"[7]"},{"why":"Soleymani et al.'s URLLC MIMO optimization supplies the finite-block-length rate expression and its concave lower bound (Lemma 3) used throughout the applications.","marker":"[18]"},{"why":"Aubry et al.'s sequential optimization lemma states the three MM surrogate conditions that the paper relies on for convergence to a stationary point.","marker":"[58]"},{"why":"Sun, Babu, and Palomar's MM survey provides the convex-concave procedure and first-order lower bounds used to convexify non-convex constraints.","marker":"[59]"},{"why":"Soleymani et al.'s RSMA work is the source of Lemma 11, the complex quadratic lower bound used to handle non-convex RIS coefficient feasibility sets.","marker":"[57]"},{"why":"Dinkelbach's classical single-ratio fractional programming algorithm is the origin of the solver family that this framework generalizes beyond.","marker":"[6]"}],"fun_headline_variants":["Single-loop solver for sums and products of fractional functions","Fractional matrix programming: one loop instead of two","Sum and product fractional programs: single-loop solution","Beyond Dinkelbach: single-loop FMP for sums and products","MU-MIMO fractional metrics optimized with single-loop FMP"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Single-loop solver for sums and products of fractional functions","Fractional matrix programming: one loop instead of two","Sum and product fractional programs: single-loop solution","Beyond Dinkelbach: single-loop FMP for sums and products","MU-MIMO fractional metrics optimized with single-loop FMP"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000421,"raw_usage":{"total_tokens":2249,"prompt_tokens":1114,"completion_tokens":1135,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":730,"completion_tokens_details":{"reasoning_tokens":1056}},"tokens_in":730,"tokens_out":1135,"duration_ms":10731,"temperature":1.0,"reasoning_tokens":1056,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T16:41:30.851088+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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).","supporting_citations":[{"cited_title":"Fractional programming for communication systems–Part I: Power control and beamforming,","cited_arxiv_id":null,"evidence_quote":"Shen and Yu's quadratic transform is the main baseline for maximization FMP; the paper contrasts its alternating-optimization structure with its own single-loop surrogate approach."},{"cited_title":"Algorithms for generalized fractional programming,","cited_arxiv_id":null,"evidence_quote":"Crouzeix and Ferland's generalized Dinkelbach algorithm is the twin-loop baseline for multiple-ratio max-min FMP that the framework compares against."},{"cited_title":"Optimization of the downlink spectral-and energy-efficiency of RIS- aided multi-user URLLC MIMO systems,","cited_arxiv_id":null,"evidence_quote":"Soleymani et al.'s URLLC MIMO optimization supplies the finite-block-length rate expression and its concave lower bound (Lemma 3) used throughout the applications."},{"cited_title":"A new sequential optimization procedure and its applications to resource allocation for wireless systems,","cited_arxiv_id":null,"evidence_quote":"Aubry et al.'s sequential optimization lemma states the three MM surrogate conditions that the paper relies on for convergence to a stationary point."},{"cited_title":"Majorization-minimization algo- rithms in signal processing, communications, and machine learning,","cited_arxiv_id":null,"evidence_quote":"Sun, Babu, and Palomar's MM survey provides the convex-concave procedure and first-order lower bounds used to convexify non-convex constraints."},{"cited_title":"Rate splitting in MIMO RIS-assisted systems with hardware impairments and improper signaling,","cited_arxiv_id":null,"evidence_quote":"Soleymani et al.'s RSMA work is the source of Lemma 11, the complex quadratic lower bound used to handle non-convex RIS coefficient feasibility sets."},{"cited_title":"On nonlinear fractional programming,","cited_arxiv_id":null,"evidence_quote":"Dinkelbach's classical single-ratio fractional programming algorithm is the origin of the solver family that this framework generalizes beyond."}],"review_version":1}