Lagrange Dual Transform-Based Fractional Programming

• LDTFP is a framework that uses the Lagrange dual transform to remove logarithms from ratio objectives, facilitating nonconvex optimization.
• It is applied in wireless communications for tasks like sum-rate and energy-efficiency maximization using techniques such as quadratic transform and MM methods.
• Algorithmic implementations of LDTFP yield closed-form iterative updates and enhanced convergence compared to traditional manifold-based approaches.

Searching arXiv for foundational and papers on Lagrangian dual transform and fractional programming. arxiv_search(query="Lagrangian dual transform fractional programming Shen Yu 2018", max_results=5) arxiv_search(query="fractional programming communication systems part I quadratic transform Shen Yu arXiv", max_results=10) Lagrange Dual Transform-Based Fractional Programming (LDTFP) denotes a class of fractional-programming constructions in which the Lagrangian dual transform (LDT) is used to remove logarithms from ratio-valued objectives, typically of the form $\log(1+\mathrm{SINR})$ or more general log-ratio sums, and is often followed by the quadratic transform (QT) or embedded in broader alternating-optimization frameworks. Within the cited literature, this methodology appears in sum-rate maximization for reciprocal beyond-diagonal reconfigurable intelligent surfaces (BD-RIS), in energy-efficiency maximization for tri-hybrid beamforming with radiation-center reconfigurable antenna arrays (RCRAA), and in generalized mixed max-and-min log-ratio programs for wireless networks; more recent work also re-examines the classical LDT+QT construction from a minorization-maximization (MM) perspective and proposes a tighter surrogate alternative (Fidanovski et al., 10 Nov 2025, Li et al., 21 Aug 2025, Chen et al., 2023, Jiao et al., 2 Jul 2026).

1. Problem classes and optimization structure

LDTFP is used for nonconvex objectives in which logarithms are composed with ratios. A generic form considered in the literature is

$$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$

with $A_m(x)\ge 0$ and $B_m(x)>0$. In coordinated uplink scheduling and power control, this becomes a weighted sum-rate (WSR) maximization over mixed discrete-continuous variables, where each base station schedules at most one user and the resulting $\mathrm{SINR}_i$ enters $\omega_{s_i}\log(1+\mathrm{SINR}_i)$ (Jiao et al., 2 Jul 2026).

A broader formulation appears in mixed max-and-min fractional programming. There, the objective contains both positive and negative logarithmic ratio terms,

$$\sum_{n=1}^{N_0} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr) -\sum_{n=N_0+1}^{N} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr),$$

under assumptions that align the concavity or convexity of $A_n$ and $B_n$ with the sign of each term. This generalization is motivated by settings such as secure data-rate maximization, age-of-information minimization, and Cramer-Rao bound minimization (Chen et al., 2023).

Two application-specific formulations illustrate how LDTFP is specialized. In reciprocal BD-RIS scattering-matrix design, the optimization variable is the reciprocal scattering matrix $\mathbf\Theta$ under symmetry and unitarity constraints, and the objective is

$$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$0

where

$$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$1

The feasible set enforces $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$2 and $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$3, or block-unitarity in the group-connected case (Fidanovski et al., 10 Nov 2025).

In tri-hybrid beamforming for RCRAA, the original problem is a single-ratio energy-efficiency maximization: $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$4 with $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$5, $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$6, and constant-modulus analog constraints (Li et al., 21 Aug 2025).

Setting	Objective	Role of LDTFP
Multicell uplink scheduling	WSR maximization	Classical LDT+QT creates separable surrogates
Mixed max-and-min FP	Sum of positive and negative log-ratios	Generalized LDT yields an MM minorizer
Reciprocal BD-RIS design	Sum-rate maximization over $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$7	LDT+QT supports manifold optimization
Tri-hybrid beamforming	Energy-efficiency maximization	LDTFP gives closed-form iterative updates

These formulations show that LDTFP is not tied to a single architecture or variable type. It has been used with continuous beamformers, discrete scheduling variables, manifold-constrained scattering matrices, and outer single-ratio programs.

2. Core Lagrangian dual transform

The central LDT identity rewrites a log-ratio term by introducing an auxiliary nonnegative variable. For the generic sum-log-ratio problem,

$$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$8

and for fixed $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$9 the optimal auxiliary variable is

$A_m(x)\ge 0$0

This reformulation moves each ratio outside the logarithm without changing the objective value (Jiao et al., 2 Jul 2026).

In the BD-RIS formulation, the same construction is written with base-2 logarithms and userwise auxiliary variables $A_m(x)\ge 0$1: $A_m(x)\ge 0$2 For fixed $A_m(x)\ge 0$3, the optimum is $A_m(x)\ge 0$4, and summing the resulting surrogates while adding a symmetry penalty yields

$A_m(x)\ge 0$5

The paper explicitly states that the identity is from Shen–Yu 2018 (Fidanovski et al., 10 Nov 2025).

The generalized LDT in mixed max-and-min fractional programming separates positive and negative log-ratio terms by introducing $A_m(x)\ge 0$6 for the “max” group and $A_m(x)\ge 0$7 for the “min” group. At the current iterate $A_m(x)\ge 0$8, the resulting surrogate is

$A_m(x)\ge 0$9

$B_m(x)>0$0

where $B_m(x)>0$1 and $B_m(x)>0$2. By construction, $B_m(x)>0$3 and $B_m(x)>0$4 for all $B_m(x)>0$5 (Chen et al., 2023).

The main algorithmic significance of LDT is therefore structural rather than merely algebraic: it replaces a logarithm of a ratio by a surrogate containing a pure ratio. That shift is what enables the next transformation stage.

3. Combination with quadratic transform and outer fractional methods

After LDT, the remaining nonconvexity is usually a pure fractional term. The standard next step is the quadratic transform. In the generic real-valued form,

$B_m(x)>0$6

while in complex-valued beamforming problems the auxiliary variable is complex and the transformed term takes the form

$B_m(x)>0$7

At the optimum,

$B_m(x)>0$8

This produces a doubly-surrogate objective in $B_m(x)>0$9 for BD-RIS design (Fidanovski et al., 10 Nov 2025).

In the energy-efficiency problem for tri-hybrid beamforming, LDTFP is not used in isolation. The outer single-ratio objective is first handled by Dinkelbach’s transform with parameter $\mathrm{SINR}_i$0, the coupling constraint $\mathrm{SINR}_i$1 is enforced by penalty dual decomposition with penalty weight $\mathrm{SINR}_i$2, and the spectral-efficiency numerator is rewritten by an epigraph variable $\mathrm{SINR}_i$3. LDT is then applied to the constraints $\mathrm{SINR}_i$4, producing

$\mathrm{SINR}_i$5

followed by QT with a complex scalar $\mathrm{SINR}_i$6 for each user. The fully decoupled objective is

$\mathrm{SINR}_i$7

with block-coordinate updates for $\mathrm{SINR}_i$8, $\mathrm{SINR}_i$9, $\omega_{s_i}\log(1+\mathrm{SINR}_i)$0, the vectorized beamformer $\omega_{s_i}\log(1+\mathrm{SINR}_i)$1, $\omega_{s_i}\log(1+\mathrm{SINR}_i)$2, and $\omega_{s_i}\log(1+\mathrm{SINR}_i)$3. The update of $\omega_{s_i}\log(1+\mathrm{SINR}_i)$4 is obtained from the linear system $\omega_{s_i}\log(1+\mathrm{SINR}_i)$5, while $\omega_{s_i}\log(1+\mathrm{SINR}_i)$6 uses phase projection of $\omega_{s_i}\log(1+\mathrm{SINR}_i)$7 and $\omega_{s_i}\log(1+\mathrm{SINR}_i)$8 uses least-squares or SVD-based normalization (Li et al., 21 Aug 2025).

A recurrent misconception is that LDT by itself solves the entire nonconvex problem. The literature instead treats LDT as a front-end transform that removes the logarithm but usually leaves a pure ratio, a block-separable surrogate, or a constrained subproblem that is handled by QT, MM, a convex subsolver, or a manifold method. The cited papers are consistent on this point.

4. Algorithmic realizations

In reciprocal BD-RIS design, the transformed problem is solved by alternating updates embedded in a Riemannian conjugate-gradient algorithm on the product Stiefel manifold. The procedure initializes $\omega_{s_i}\log(1+\mathrm{SINR}_i)$9 on the product Stiefel manifold as block-unitary and random symmetric, along with auxiliary $$\sum_{n=1}^{N_0} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr) -\sum_{n=N_0+1}^{N} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr),$$0 and $$\sum_{n=1}^{N_0} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr) -\sum_{n=N_0+1}^{N} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr),$$1. At each iteration it updates

$$\sum_{n=1}^{N_0} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr) -\sum_{n=N_0+1}^{N} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr),$$2

computes the closed-form Euclidean gradient of $$\sum_{n=1}^{N_0} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr) -\sum_{n=N_0+1}^{N} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr),$$3, subtracts the penalty term $$\sum_{n=1}^{N_0} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr) -\sum_{n=N_0+1}^{N} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr),$$4 on each block, projects the result onto the tangent space,

$$\sum_{n=1}^{N_0} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr) -\sum_{n=N_0+1}^{N} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr),$$5

and then uses the Polak-Ribière rule, Armijo line search, and a QR-based retraction

$$\sum_{n=1}^{N_0} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr) -\sum_{n=N_0+1}^{N} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr),$$6

Termination occurs when $$\sum_{n=1}^{N_0} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr) -\sum_{n=N_0+1}^{N} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr),$$7, and each block is finally projected to exact symmetry and unitarity by

$$\sum_{n=1}^{N_0} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr) -\sum_{n=N_0+1}^{N} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr),$$8

The paper states that LDT and QT reduce the complexity of the optimization problem for the scattering matrix solution while providing notable performance gains compared to state-of-the-art methods under the same system conditions (Fidanovski et al., 10 Nov 2025).

In the mixed max-and-min framework, the algorithm is MM-style rather than manifold-based. Starting from $$\sum_{n=1}^{N_0} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr) -\sum_{n=N_0+1}^{N} w_n\log\!\Bigl(1+\frac{A_n(x)}{B_n(x)}\Bigr),$$9, each iteration computes the current auxiliary ratios $A_n$0 and $A_n$1 and then solves the concave surrogate maximization

$A_n$2

The monotonicity chain

$A_n$3

is given explicitly (Chen et al., 2023).

In multicell uplink scheduling and power control, the classical LDT+QT construction leads to a closed-form FP surrogate $A_n$4 whose updates in $A_n$5 admit closed forms. With $A_n$6 fixed, the problem decouples across blocks of $A_n$7, enabling separate scheduling updates even when $A_n$8 is partly discrete. This separability is central to the usefulness of LDTFP in discrete-continuous WSR programs (Jiao et al., 2 Jul 2026).

These realizations indicate that LDTFP is best understood as a transform layer inside larger algorithmic systems rather than as a single fixed solver.

5. Complexity and reported empirical behavior

The complexity of LDTFP depends strongly on the downstream solver. In the reciprocal BD-RIS problem, the reported per-iteration cost is

$A_n$9

The individual components are also stated explicitly: building all $B_n$0 costs $B_n$1, computing $B_n$2 and $B_n$3 for all $B_n$4 costs $B_n$5, and the gradient double sums cost $B_n$6. This is reported to be of the same order as the state-of-the-art manifold-based solver in Fidanovski et al. 2025, but with simpler gradient expressions and the overhead $B_n$7 for the FP updates (Fidanovski et al., 10 Nov 2025).

The same paper reports convergence-speed comparisons in Fig. 4: single-connected $B_n$8 iterations versus $B_n$9 iterations for the state-of-the-art method, Group(2) $\mathbf\Theta$0 versus $\mathbf\Theta$1, Group(4) $\mathbf\Theta$2 versus $\mathbf\Theta$3, and fully-connected $\mathbf\Theta$4 versus $\mathbf\Theta$5. Under uniform power allocation, the LDTFP-based design consistently outperforms Fidanovski et al. 2025 by $\mathbf\Theta$6–$\mathbf\Theta$7 in absolute rate across SNR, outperforms Yahya et al. 2024 in the single-connected case, yields right-shifted CDF distributions for every architecture, and exhibits a rate gap versus the state of the art that widens as the number of reflecting elements $\mathbf\Theta$8 increases (Fidanovski et al., 10 Nov 2025).

In the tri-hybrid beamforming problem, the dominant cost of LDTFP is the solution of the linear system for $\mathbf\Theta$9, giving per-inner-iteration complexity $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$00. The same source contrasts this with the DQTFP scheme, which requires solving a convex program over $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$01 via a general-purpose solver and is described as having worst-case complexity typically $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$02 or worse. In the reported MATLAB implementation with $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$03 and $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$04, LDTFP took $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$05 per iteration while DQTFP took $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$06. The abstract summarizes the tradeoff as a significant reduction in computational complexity with only minor performance loss (Li et al., 21 Aug 2025).

These results do not imply a universal complexity ranking for all LDT-based methods. A plausible implication is that LDTFP is most attractive when the transformed subproblems admit closed forms, linear systems, or efficient manifold steps; the cited papers repeatedly exploit exactly that structure.

6. MM interpretation, scope relative to QT, and later surrogate refinements

The MM interpretation is central to the theoretical status of LDTFP. In the mixed max-and-min framework, the LDT-derived surrogate $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$07 is an exact minorizer: it touches the objective at the current iterate and lower-bounds it elsewhere. Under smoothness assumptions and Slater-type conditions for $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$08, standard MM results are invoked to conclude that $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$09 is non-decreasing and bounded above, hence convergent, and that every limit point is a stationary point of $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$10 on $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$11 (Chen et al., 2023).

That same source makes a precise distinction between LDT and QT. QT applies to sums of pure ratios $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$12 and is described as very general, handling sum of positive ratios, max-FP, min-FP, and mixed FP; however, it does not remove the logarithm when the objective is $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$13. LDT, by contrast, applies when the outer function is logarithmic, moves the ratio outside the log via epigraph and dualization, and leaves no log in the $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$14-subproblem, but it must specialize to the log case. The same paper explicitly notes that one can combine LDT and QT in practice (Chen et al., 2023).

A later reassessment argues that the classical LDT surrogate is conservative because of its reciprocal-coordinate construction. For a single ratio $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$15 with $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$16, the LDT lower bound $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$17 is reported to satisfy three properties: $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$18, local curvature $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$19, and bounded asymptotic behavior $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$20 as $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$21 while $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$22. On that basis, the paper proposes the reciprocal-inversion transform (RIT), yielding

$$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$23

with $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$24 and $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$25. The stated properties are $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$26, first-order tightness at $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$27, and unbounded growth as $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$28 (Jiao et al., 2 Jul 2026).

The resulting surrogate-enhanced FP (SEFP) algorithm remains compatible with QT, retains per-cell separability, and admits closed-form updates for auxiliary variables, scheduling decisions, and transmit powers. In the reported 7-cell wrap-around Rayleigh simulations, proportional-fairness utility improved from about $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$29 for classical FP to $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$30 for SEFP with $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$31 confidence interval $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$32 gain; equal-weight sum-rate mean rose from $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$33 to $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$34, with SEFP winning in approximately $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$35 of realizations; random-priority WSR mean rose from $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$36 to $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$37, winning in approximately $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$38 of tests; and across SNRs from $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$39 to $$\max_{x\in\mathcal X}\;F(x)\triangleq \sum_{m=1}^M\omega_m\log\Bigl(1+\tfrac{A_m(x)}{B_m(x)}\Bigr),$$40, SEFP preserved a uniform proportional-fairness utility advantage (Jiao et al., 2 Jul 2026).

Taken together, these works place LDTFP in a precise methodological niche. It is a logarithm-specific transform framework that often serves as the first stage of a larger surrogate-based algorithm, has a clean MM interpretation in generalized settings, and remains an active target for refinement when tighter lower bounds can preserve the separability and closed-form updates that made the classical LDT+QT pipeline effective.