---
title: Lagrange Dual Transform-Based Fractional Programming
url: https://www.emergentmind.com/topics/lagrange-dual-transform-based-fractional-programming-ldtfp
type: topic
---

# Lagrange Dual Transform-Based Fractional Programming

Searching arXiv for foundational and recent papers on Lagrangian dual transform and fractional programming.
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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 [2511.07683] [2508.15924] [2305.02704] [2607.01769].

## 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 $\omega_{s_i}\log(1+\mathrm{SINR}_i)$ [2607.01769].

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 [2305.02704].

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
\[
\max_{\mathbf\Theta}\;\sum_{k=1}^K \log_2\bigl(1+\gamma_k(\mathbf\Theta)\bigr),
\]
where
\[
\gamma_k(\mathbf\Theta)=
\frac{\bigl|\mathbf h_k^T\mathbf\Theta\mathbf H_{\rm TX}\mathbf v_k\bigr|^2}
{\sum_{i\neq k}\bigl|\mathbf h_k^T\mathbf\Theta\mathbf H_{\rm TX}\mathbf v_i\bigr|^2+N_0}.
\]
The feasible set enforces $\mathbf\Theta=\mathbf\Theta^T$ and $\mathbf\Theta\mathbf\Theta^H=\mathbf I$, or block-unitarity in the group-connected case [2511.07683].

In tri-hybrid beamforming for RCRAA, the original problem is a single-ratio energy-efficiency maximization:
\[
\max_{F_{\rm RF},F_{\rm BB}}
\frac{R_{\rm SE}(F)}
{\|F\|_F^2/\eta_{\rm PA}+P_C}
\quad\text{s.t.}\quad F_{\rm RF}\in\mathcal C,\;\|F\|_F^2\le P_{\max},
\]
with $F=F_{\rm RF}F_{\rm BB}$, $R_{\rm SE}(F)=\sum_{k=1}^K\log_2(1+\gamma_k(F))$, and constant-modulus analog constraints [2508.15924].

| 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 $\mathbf\Theta$ | 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,
\[
F(x)=\max_{\gamma\ge 0}
\sum_{m=1}^M \omega_m\Bigl[
\log(1+\gamma_m)-\gamma_m
+\frac{(1+\gamma_m)A_m(x)}{A_m(x)+B_m(x)}
\Bigr],
\]
and for fixed $x$ the optimal auxiliary variable is
\[
\gamma_m^\star=\frac{A_m(x)}{B_m(x)}.
\]
This reformulation moves each ratio outside the logarithm without changing the objective value [2607.01769].

In the BD-RIS formulation, the same construction is written with base-2 logarithms and userwise auxiliary variables $\tau_k\ge 0$:
\[
\log_2(1+\gamma_k)
=\max_{\tau_k\ge 0}\;
\Bigl[
\log_2(1+\tau_k)-\frac{\tau_k}{\ln 2}
+\frac{1+\tau_k}{\ln 2}
\frac{|\mathbf h_k^T\mathbf\Theta\mathbf H_{\rm TX}\mathbf v_k|^2}
{\sum_i |\mathbf h_k^T\mathbf\Theta\mathbf H_{\rm TX}\mathbf v_i|^2+N_0}
\Bigr].
\]
For fixed $\mathbf\Theta$, the optimum is $\tau_k=\gamma_k(\mathbf\Theta)$, and summing the resulting surrogates while adding a symmetry penalty yields
\[
\sum_{k=1}^K\overline\eta_k(\tau_k)-\nu\|\mathbf\Theta-\mathbf\Theta^T\|_F^2.
\]
The paper explicitly states that the identity is from Shen–Yu 2018 [2511.07683].

The generalized LDT in mixed max-and-min fractional programming separates positive and negative log-ratio terms by introducing $\gamma_n\ge 0$ for the “max” group and $\tilde v_n\in(0,1)$ for the “min” group. At the current iterate $\hat x$, the resulting surrogate is
\[
g(x|\hat x)
= \sum_{n=1}^{N_0}\Bigl[
w_n\log(1+\Gamma_n)-w_n\Gamma_n
+w_n(1+\Gamma_n)\frac{A_n(x)}{A_n(x)+B_n(x)}
\Bigr]
\]
\[
+\sum_{n=N_0+1}^{N}\Bigl[
w_n\log(1-\tilde\Gamma_n)+w_n\tilde\Gamma_n
-w_n(1-\tilde\Gamma_n)\frac{A_n(x)}{B_n(x)}
\Bigr],
\]
where $\Gamma_n=A_n(\hat x)/B_n(\hat x)$ and $\tilde\Gamma_n=A_n(\hat x)/[A_n(\hat x)+B_n(\hat x)]$. By construction, $g(\hat x|\hat x)=F(\hat x)$ and $g(x|\hat x)\le F(x)$ for all $x$ [2305.02704].

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,
\[
\frac{A}{B}=\max_y \bigl[2y\sqrt{A}-y^2B\bigr],
\]
while in complex-valued beamforming problems the auxiliary variable is complex and the transformed term takes the form
\[
\max_{y_k}\;
\Bigl[
2\Re\{y_k^*(\mathbf e_k\mathbf v_k)\}
-|y_k|^2\bigl(\sum_i|\mathbf e_k\mathbf v_i|^2+N_0\bigr)
\Bigr].
\]
At the optimum,
\[
y_k=\frac{\mathbf e_k\mathbf v_k}{\sum_i|\mathbf e_k\mathbf v_i|^2+N_0}.
\]
This produces a doubly-surrogate objective in $(\mathbf\Theta,\{\tau_k\},\{y_k\})$ for BD-RIS design [2511.07683].

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 $\omega$, the coupling constraint $F=F_{\rm RF}F_{\rm BB}$ is enforced by penalty dual decomposition with penalty weight $\mu_2$, and the spectral-efficiency numerator is rewritten by an epigraph variable $t$. LDT is then applied to the constraints $t_k\le \gamma_k(F)$, producing
\[
\lambda_k^*=\frac{1}{(1+\gamma_k(F))\ln 2},
\]
followed by QT with a complex scalar $z_k$ for each user. The fully decoupled objective is
\[
G_{lq}(f,F_{\rm RF},F_{\rm BB},t,z;\omega,\mu_2)
\]
with block-coordinate updates for $\omega$, $t_k$, $z_k$, the vectorized beamformer $f=\mathrm{vec}(F)$, $F_{\rm RF}$, and $F_{\rm BB}$. The update of $f$ is obtained from the linear system $\Psi f=q$, while $F_{\rm RF}$ uses phase projection of $F F_{\rm BB}^H$ and $F_{\rm BB}$ uses least-squares or SVD-based normalization [2508.15924].

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 $\mathbf\Theta^{(0)}$ on the product Stiefel manifold as block-unitary and random symmetric, along with auxiliary $\tau_k^{(0)}$ and $y_k^{(0)}$. At each iteration it updates
\[
\tau_k^{(i)}=\gamma_k(\mathbf\Theta^{(i)}),\qquad
y_k^{(i)}=\frac{\mathbf e_k^{(i)}\mathbf v_k}{\sum_i|\mathbf e_k^{(i)}\mathbf v_i|^2+N_0},
\]
computes the closed-form Euclidean gradient of $\sum_k\hat\eta_k$, subtracts the penalty term $4\nu(\mathbf\Theta_g-\mathbf\Theta_g^T)$ on each block, projects the result onto the tangent space,
\[
\mathrm T_{\mathbf\Theta_g}(G)
=G-\frac12\bigl[\mathbf\Theta_g(\mathbf\Theta_g^H G)+(G^H\mathbf\Theta_g)\mathbf\Theta_g\bigr],
\]
and then uses the Polak-Ribière rule, Armijo line search, and a QR-based retraction
\[
\mathbf\Theta_g^{(i+1)}
=\mathrm R_{\mathbf\Theta_g}\bigl(\mathbf\Theta_g^{(i)},\alpha^{(i)}\Xi^{(i)}\bigr)
=\mathcal Q(\mathbf\Theta_g^{(i)}+\alpha\Xi).
\]
Termination occurs when $|\eta(\mathbf\Theta^{(i+1)})-\eta(\mathbf\Theta^{(i)})|<\epsilon$, and each block is finally projected to exact symmetry and unitarity by
\[
\mathbf\Theta_g\leftarrow \mathrm{SVD\_proj}\bigl((\mathbf\Theta_g+\mathbf\Theta_g^T)/2\bigr).
\]
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 [2511.07683].

In the mixed max-and-min framework, the algorithm is MM-style rather than manifold-based. Starting from $x^{(0)}\in X$, each iteration computes the current auxiliary ratios $\Gamma_n$ and $\tilde\Gamma_n$ and then solves the concave surrogate maximization
\[
x^{(t+1)}=\arg\max_{x\in X} g(x|x^{(t)}).
\]
The monotonicity chain
\[
F(x^{(t+1)})\ge g(x^{(t+1)}|x^{(t)})\ge g(x^{(t)}|x^{(t)})=F(x^{(t)})
\]
is given explicitly [2305.02704].

In multicell uplink scheduling and power control, the classical LDT+QT construction leads to a closed-form FP surrogate $F_{\ell q}(x,\gamma,y)$ whose updates in $\{\gamma,y\}$ admit closed forms. With $\{\gamma,y\}$ fixed, the problem decouples across blocks of $x$, enabling separate scheduling updates even when $x$ is partly discrete. This separability is central to the usefulness of LDTFP in discrete-continuous WSR programs [2607.01769].

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
\[
\mathcal O\bigl(KNR + K^2N + K^2\,G\,(R/G)^2\bigr)
\approx
\mathcal O\bigl(K^2G(R/G)^2 + K^2N\bigr).
\]
The individual components are also stated explicitly: building all $\mathbf e_k=\mathbf h_k^T\mathbf\Theta\mathbf H_{\rm TX}$ costs $\mathcal O(KNR)$, computing $\tau_k$ and $y_k$ for all $k$ costs $\mathcal O(K^2N)$, and the gradient double sums cost $\mathcal O(K^2G(R/G)^2)$. 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 $\mathcal O(K^2N)$ for the FP updates [2511.07683].

The same paper reports convergence-speed comparisons in Fig. 4: single-connected $\approx 50$ iterations versus $50$ iterations for the state-of-the-art method, Group(2) $\approx 100$ versus $700$, Group(4) $\approx 400$ versus $1\,000$, and fully-connected $\approx 700$ versus $2\,500$. Under uniform power allocation, the LDTFP-based design consistently outperforms Fidanovski et al. 2025 by $\approx 5$–$15\%$ 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 $R$ increases [2511.07683].

In the tri-hybrid beamforming problem, the dominant cost of LDTFP is the solution of the linear system for $f$, giving per-inner-iteration complexity $\simeq \mathcal O(N^3)=\mathcal O((N_TK)^3)$. The same source contrasts this with the DQTFP scheme, which requires solving a convex program over $f$ via a general-purpose solver and is described as having worst-case complexity typically $\mathcal O(N^6)$ or worse. In the reported MATLAB implementation with $N_T=8$ and $K=4$, LDTFP took $\approx 0.007\,\mathrm{s}$ per iteration while DQTFP took $\approx 12\,\mathrm{s}$. The abstract summarizes the tradeoff as a significant reduction in computational complexity with only minor performance loss [2508.15924].

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 $g(x|\hat x)$ 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 $X$, standard MM results are invoked to conclude that $\{F(x^{(t)})\}$ is non-decreasing and bounded above, hence convergent, and that every limit point is a stationary point of $F$ on $X$ [2305.02704].

That same source makes a precise distinction between LDT and QT. QT applies to sums of pure ratios $\sum A_n/B_n$ 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 $f(A_n/B_n)=\log(1\pm A_n/B_n)$. 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 $x$-subproblem, but it must specialize to the log case. The same paper explicitly notes that one can combine LDT and QT in practice [2305.02704].

A later reassessment argues that the classical LDT surrogate is conservative because of its reciprocal-coordinate construction. For a single ratio $r=A/B$ with $f(r)=\log(1+r)$, the LDT lower bound $\ell_{\mathrm{LDT}}(r;\bar r)$ is reported to satisfy three properties: $\ell_{\mathrm{LDT}}(0;\bar r)=\log(1+\bar r)-\bar r<0=f(0)$, local curvature $\ell''(r)|_{r=\bar r}=2f''(\bar r)$, and bounded asymptotic behavior $\ell_{\mathrm{LDT}}\to \log(1+\bar r)+1<\infty$ as $r\to\infty$ while $f(r)\to\infty$. On that basis, the paper proposes the reciprocal-inversion transform (RIT), yielding
\[
f(r)\ge \ell_{\mathrm{RIT}}(r;\bar r)
=\frac{c(\bar r)\,r}{\bar r^2+b(\bar r)\,r},
\]
with $b(v)=(1+v)\log(1+v)-v$ and $c(v)=(1+v)\log^2(1+v)$. The stated properties are $\ell_{\mathrm{RIT}}(0;\bar r)=0=f(0)$, first-order tightness at $r=\bar r$, and unbounded growth as $r\to\infty$ [2607.01769].

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 $65.9$ for classical FP to $68.9$ for SEFP with $95\%$ confidence interval $[2.79,3.20]$ gain; equal-weight sum-rate mean rose from $627.7$ to $645.1$, with SEFP winning in approximately $96\%$ of realizations; random-priority WSR mean rose from $826.5$ to $853.0$, winning in approximately $78\%$ of tests; and across SNRs from $0$ to $14\,\mathrm{dB}$, SEFP preserved a uniform proportional-fairness utility advantage [2607.01769].

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.

Source: https://www.emergentmind.com/topics/lagrange-dual-transform-based-fractional-programming-ldtfp