Dual Quadratic Transform Fractional Programming

• DQTFP is a fractional programming paradigm that decouples ratio terms using dual and quadratic transforms for tractable subproblems.
• It applies Lagrangian dual, quadratic, and inverse quadratic transforms to diverse settings such as beamforming, clustering, and robust optimization.
• The method achieves closed-form auxiliary updates with alternating optimization and convergence guarantees, enhancing efficiency in wireless and sensing applications.

Dual Quadratic Transform-Based Fractional Programming (DQTFP) denotes a family of auxiliary-variable methods for fractional programs in which ratio terms are converted into equivalent or majorization–minimization surrogates that decouple numerators and denominators. The cited literature suggests that the label does not refer to a single universally fixed construction: in some works it means a Lagrangian dual transform followed by the quadratic transform for concave functions of ratios, in others it means a unified pair of quadratic and inverse-quadratic transforms for mixed maximization–minimization, and in still others it denotes a quadratic-transform stage combined with a duality-based solution of an inner nonconvex subproblem or a second quadratic transform applied to nested ratios. Across these variants, the common objective is to obtain closed-form auxiliary updates and tractable primal subproblems—typically convex quadratic programs, semidefinite programs, linear programs, or manifold-constrained updates—for applications in wireless communications, sensing, graph clustering, and machine learning (Shen et al., 13 Mar 2025, Chen et al., 2023, Iimori et al., 2020).

1. Terminology and scope

The cited literature suggests that the phrase “DQTFP” is best understood as a technical family rather than a single canonical algorithm. In one line of work, it is explicitly a two-stage procedure that applies a Lagrangian dual transform to a concave function of a ratio and then a quadratic transform to the resulting fractional term. In another, it is a unified framework that combines the standard quadratic transform for maximization with an inverse quadratic transform for minimization and mixed max–min objectives. In robust beamforming, the “dual” step is not a Lagrangian dual transform for logarithms, but rather a duality-based closed-form solution of a trust-region-type QCQP in the uncertainty variables. In energy-efficiency design for tri-hybrid beamforming, “dual quadratic transform” refers to applying the quadratic transform twice: once to the outer energy-efficiency ratio and once to the inner SINR ratios (Shen et al., 13 Mar 2025, Chen et al., 2023, Iimori et al., 2020, Li et al., 21 Aug 2025).

Interpretation in the literature	Second-stage “dual” mechanism	Representative setting
Two-stage transform for concave functions of ratios	Lagrangian dual transform followed by quadratic transform	Sum-rate and matrix-ratio optimization
Unified max–min fractional programming	Inverse quadratic transform for minimization, plus quadratic transform for maximization	AoI, CRB minimization, secure rate
Robust constrained design	Lagrangian duality for an inner worst-case QCQP	Imperfect-CSI PD-NOMA beamforming
Nested-ratio handling	Second quadratic transform applied to outer and inner ratios	Energy-efficiency maximization in THBF

A recurrent source of confusion is the word “dual.” In the LDT+QT literature it refers to the Lagrangian dual transform that removes a concave outer function such as $\log(1+\gamma)$. In the mixed max–min literature it refers instead to the inverse quadratic transform that majorizes a minimization ratio. In robust optimization papers, it refers to strong duality of an inner nonconvex QCQP. These are related only at the level of purpose—ratio decoupling and tractable alternating optimization—not at the level of a unique algebraic identity.

2. Core transform identities

The standard quadratic transform starts from the scalar identity

$$\frac{u}{v}=\max_{y\in\mathbb{R}}\left(2y\sqrt{u}-y^2 v\right),\qquad y^\star=\frac{\sqrt{u}}{v},$$

for $u\ge 0$ and $v>0$. In complex beamforming form,

$$\frac{|h^H w|^2}{B(x)} = \max_{y\in\mathbb{C}} \left(2\operatorname{Re}\{y^\ast h^H w\}-|y|^2 B(x)\right), \qquad y^\star=\frac{h^H w}{B(x)}.$$

For matrix ratios, one similarly has

$$\operatorname{Tr}\!\big(S(x)^H B(x)^{-1}S(x)\big) = \max_Y \left(2\operatorname{Re}\operatorname{Tr}(Y^H S(x))-\operatorname{Tr}(Y^H B(x)Y)\right), \qquad Y^\star=B(x)^{-1}S(x).$$

These identities are exact at the auxiliary optimum and convert nonlinear couplings between numerator and denominator into bilinear and quadratic terms (Shen et al., 13 Mar 2025).

For concave functions of ratios, the Lagrangian dual transform moves the ratio outside the outer nonlinearity. For $f(\gamma)=\log(1+\gamma)$,

$$\log(1+\gamma) = \max_{u\ge 0} \left[ \log(1+u)-u+(1+u)\frac{\gamma}{1+\gamma} \right], \qquad u^\star=\gamma.$$

The remaining fractional term is then handled by the quadratic transform. This is the algebraic core of the LDT+QT formulation for weighted sum-rate and related objectives (Shen et al., 13 Mar 2025).

For minimization, the unified mixed max–and–min framework introduces the inverse quadratic transform

$$\frac{u}{v} = \min_{t\in\mathbb{R}} \frac{1}{[2t\sqrt{v}-t^2u]_+}, \qquad t^\star=\frac{\sqrt{v}}{u},$$

with $[a]_+=\lim_{\epsilon\to 0^+}\max\{a,\epsilon\}$. This identity generates a majorizer rather than a minorizer, enabling minimization of sums of ratios and mixed objectives containing both increasing and decreasing functions of ratios. In the mixed framework, maximization terms receive quadratic-transform auxiliaries $$\frac{u}{v}=\max_{y\in\mathbb{R}}\left(2y\sqrt{u}-y^2 v\right),\qquad y^\star=\frac{\sqrt{u}}{v},$$0, while minimization terms receive inverse-transform auxiliaries $$\frac{u}{v}=\max_{y\in\mathbb{R}}\left(2y\sqrt{u}-y^2 v\right),\qquad y^\star=\frac{\sqrt{u}}{v},$$1; the resulting surrogate admits an MM interpretation and extends to matrix ratios and logarithmic forms via a generalized Lagrangian dual transform (Chen et al., 2023).

3. Alternating optimization, MM structure, and convergence

A generic DQTFP iteration alternates between auxiliary-variable updates and a primal-variable update. In the LDT+QT setting for a sum of concave functions of ratios, the steps are: update dual variables such as $$\frac{u}{v}=\max_{y\in\mathbb{R}}\left(2y\sqrt{u}-y^2 v\right),\qquad y^\star=\frac{\sqrt{u}}{v},$$2; update quadratic auxiliaries such as

$$\frac{u}{v}=\max_{y\in\mathbb{R}}\left(2y\sqrt{u}-y^2 v\right),\qquad y^\star=\frac{\sqrt{u}}{v},$$3

and solve the resulting convex quadratic subproblem in the primal variables. In the standard sum-rate beamforming example with a sum-power constraint, the beamformer update takes the MMSE-like form

$$\frac{u}{v}=\max_{y\in\mathbb{R}}\left(2y\sqrt{u}-y^2 v\right),\qquad y^\star=\frac{\sqrt{u}}{v},$$4

where $$\frac{u}{v}=\max_{y\in\mathbb{R}}\left(2y\sqrt{u}-y^2 v\right),\qquad y^\star=\frac{\sqrt{u}}{v},$$5 enforces the power constraint. At the auxiliary optimum, the transformed objective equals the original objective, and under the listed differentiability and convexity assumptions the alternating updates monotonically improve the original objective and converge to a stationary point (Shen et al., 13 Mar 2025).

The mixed max–min framework has the same structural pattern, but with lower-bounding surrogates for maximization terms and upper-bounding surrogates for minimization terms. Its MM formulation establishes monotonic improvement and stationarity under differentiability of the component functions and convexity of the feasible set. The matrix extension preserves the same logic by replacing scalar auxiliaries with matrix auxiliaries $$\frac{u}{v}=\max_{y\in\mathbb{R}}\left(2y\sqrt{u}-y^2 v\right),\qquad y^\star=\frac{\sqrt{u}}{v},$$6 and $$\frac{u}{v}=\max_{y\in\mathbb{R}}\left(2y\sqrt{u}-y^2 v\right),\qquad y^\star=\frac{\sqrt{u}}{v},$$7 (Chen et al., 2023).

A later convergence-rate analysis connected the quadratic transform and its nonhomogeneous variant to gradient projection and WMMSE. In that analysis, conventional QT and nonhomogeneous QT both achieve local objective-value error bounds of order $$\frac{u}{v}=\max_{y\in\mathbb{R}}\left(2y\sqrt{u}-y^2 v\right),\qquad y^\star=\frac{\sqrt{u}}{v},$$8, while an accelerated version using Nesterov extrapolation achieves a bound of order $$\frac{u}{v}=\max_{y\in\mathbb{R}}\left(2y\sqrt{u}-y^2 v\right),\qquad y^\star=\frac{\sqrt{u}}{v},$$9 under a $u\ge 0$0-Lipschitz gradient assumption. The same work interprets WMMSE as a special case of the auxiliary-variable framework and uses the gradient-projection view to reduce the cost of large matrix inversions in massive MIMO and ISAC settings (Shen et al., 2023).

4. Constrained, robust, and fractionally constrained formulations

Robust PD-NOMA beamforming under imperfect CSI provides a constrained interpretation of DQTFP. The problem minimizes total transmit power subject to worst-case SINR constraints for all users that must decode a given layer. The quadratic transform is applied to each SINR ratio, introducing auxiliary variables $u\ge 0$1 and converting the ratio constraints into quadratic forms in the beamformers and CSI errors. The inner uncertainty problem then becomes a trust-region-type QCQP,

$u\ge 0$2

for which strong duality is invoked. Solving the corresponding dual SDP yields the closed-form worst-case CSI update

$u\ge 0$3

The outer beamformer update is handled by SDR. For the reported setting $u\ge 0$4, $u\ge 0$5, $u\ge 0$6, and $u\ge 0$7, the paper reports empirical rank-one recovery of approximately $u\ge 0$8, convergence within $u\ge 0$9 iterations about $v>0$0 of the time, and fewer than $v>0$1 of runs hitting the cap $v>0$2 (Iimori et al., 2020).

RIS-assisted joint sensing and communications extends the methodology to fractionally constrained fractional programming, where both the objective and the constraints contain ratios. The sensing objective is a sum of Fisher-information-type matrix ratios arising from a BCRLB reformulation, while the communication requirements are SINR ratio constraints. The method introduces objective-side auxiliaries $v>0$3 and constraint-side auxiliaries $v>0$4, applies the quadratic transform to both sides, and thereby converts the problem into a sequence of subproblems that are convex except for the RIS constant-modulus conditions. Two implementations are given: a penalty-based QCQP approach and a constant-modulus linear transform that turns the subproblems into LPs with linear complexity in the number of RIS elements. Under a mild condition on the dual optimum, the LP-based subproblem achieves global optimality. The same framework is extended to the unknown-fading case through a modified BCRLB. Reported numerical effects include an approximately $v>0$5 dB gain over using CRLB with a plugged-in mean angle, an approximately $v>0$6 dB BCRLB improvement when increasing the RIS size from $v>0$7 to $v>0$8, and an approximately $v>0$9 dB BCRLB loss when the sensing-user fading coefficient is unknown (Liu et al., 13 Mar 2025).

These constrained variants show that DQTFP is not limited to unconstrained sums of ratios. The cited results indicate that the same auxiliary-variable principle can be combined with SDR, trust-region duality, QCQP penalty methods, and LP reformulations when robustness, uncertainty sets, or ratio constraints appear explicitly.

5. Representative application domains

The broadest statement of scope appears in the 2025 review, which places quadratic-transform-based FP in wireless sum-rate maximization, sensing metrics such as Fisher-information or CRB-related ratios, normalized-cut clustering, and SVM margin formulations. In wireless communications, the framework handles SINR expressions directly. In graph clustering, it supports normalized-cut-type ratio objectives and yields an FP-based clustering procedure with auxiliary updates followed by assignment steps. In SVM settings, it decouples normalization terms from linear scores and accommodates multi-class sum- or matrix-ratio formulations (Shen et al., 13 Mar 2025).

The mixed max–and–min generalization enlarges the range of applications by accommodating decreasing functions of ratios. Three applications are emphasized: age-of-information minimization, where fractional AoI terms are minimized by the inverse quadratic transform; Cramér–Rao bound minimization for sensing, where the matrix extension is used together with Schur-complement reformulation; and secure data-rate maximization, where legitimate-user SINRs are maximized while eavesdropper SINRs are minimized. The same paper also develops a generalized Lagrangian dual transform for mixed logarithmic objectives and gives closed-form updates for the corresponding auxiliaries (Chen et al., 2023).

Low-rank convex–convex quadratic fractional programming illustrates a specialized but instructive case. With a PSD numerator $$\frac{|h^H w|^2}{B(x)} = \max_{y\in\mathbb{C}} \left(2\operatorname{Re}\{y^\ast h^H w\}-|y|^2 B(x)\right), \qquad y^\star=\frac{h^H w}{B(x)}.$$0 of low rank and a PD denominator $$\frac{|h^H w|^2}{B(x)} = \max_{y\in\mathbb{C}} \left(2\operatorname{Re}\{y^\ast h^H w\}-|y|^2 B(x)\right), \qquad y^\star=\frac{h^H w}{B(x)}.$$1, the ratio

$$\frac{|h^H w|^2}{B(x)} = \max_{y\in\mathbb{C}} \left(2\operatorname{Re}\{y^\ast h^H w\}-|y|^2 B(x)\right), \qquad y^\star=\frac{h^H w}{B(x)}.$$2

admits a low-dimensional auxiliary update $$\frac{|h^H w|^2}{B(x)} = \max_{y\in\mathbb{C}} \left(2\operatorname{Re}\{y^\ast h^H w\}-|y|^2 B(x)\right), \qquad y^\star=\frac{h^H w}{B(x)}.$$3. The cited algorithm also studies a scalar rank-one decomposition and uses region checking plus Dinkelbach verification to certify global optimality. In the reported experiments, the QT routine typically converged in $$\frac{|h^H w|^2}{B(x)} = \max_{y\in\mathbb{C}} \left(2\operatorname{Re}\{y^\ast h^H w\}-|y|^2 B(x)\right), \qquad y^\star=\frac{h^H w}{B(x)}.$$4–$$\frac{|h^H w|^2}{B(x)} = \max_{y\in\mathbb{C}} \left(2\operatorname{Re}\{y^\ast h^H w\}-|y|^2 B(x)\right), \qquad y^\star=\frac{h^H w}{B(x)}.$$5 iterations per region; in $$\frac{|h^H w|^2}{B(x)} = \max_{y\in\mathbb{C}} \left(2\operatorname{Re}\{y^\ast h^H w\}-|y|^2 B(x)\right), \qquad y^\star=\frac{h^H w}{B(x)}.$$6 tested subproblems it converged in a single iteration, and a one-iteration-per-region fast variant remained below $$\frac{|h^H w|^2}{B(x)} = \max_{y\in\mathbb{C}} \left(2\operatorname{Re}\{y^\ast h^H w\}-|y|^2 B(x)\right), \qquad y^\star=\frac{h^H w}{B(x)}.$$7 error across the tested configurations (Krishtal et al., 2023).

6. Geometry-aware implementations, computational trade-offs, and later refinements

DQTFP has also been combined with non-Euclidean optimization geometry. In reciprocal BD-RIS design, the objective is the MU-MISO sum-rate and the decision variable is a reciprocal scattering matrix $$\frac{|h^H w|^2}{B(x)} = \max_{y\in\mathbb{C}} \left(2\operatorname{Re}\{y^\ast h^H w\}-|y|^2 B(x)\right), \qquad y^\star=\frac{h^H w}{B(x)}.$$8 satisfying symmetry and unitary or block-unitary constraints. The method applies the Lagrangian dual transform and quadratic transform to obtain a smooth surrogate in $$\frac{|h^H w|^2}{B(x)} = \max_{y\in\mathbb{C}} \left(2\operatorname{Re}\{y^\ast h^H w\}-|y|^2 B(x)\right), \qquad y^\star=\frac{h^H w}{B(x)}.$$9, then performs Riemannian gradient or conjugate-gradient ascent on the Stiefel manifold, with symmetry handled by a penalty and final projection. The reported per-iteration gradient cost for the group-connected architecture is $$\operatorname{Tr}\!\big(S(x)^H B(x)^{-1}S(x)\big) = \max_Y \left(2\operatorname{Re}\operatorname{Tr}(Y^H S(x))-\operatorname{Tr}(Y^H B(x)Y)\right), \qquad Y^\star=B(x)^{-1}S(x).$$0, the QR retraction cost is $$\operatorname{Tr}\!\big(S(x)^H B(x)^{-1}S(x)\big) = \max_Y \left(2\operatorname{Re}\operatorname{Tr}(Y^H S(x))-\operatorname{Tr}(Y^H B(x)Y)\right), \qquad Y^\star=B(x)^{-1}S(x).$$1, and typical iteration counts are approximately $$\operatorname{Tr}\!\big(S(x)^H B(x)^{-1}S(x)\big) = \max_Y \left(2\operatorname{Re}\operatorname{Tr}(Y^H S(x))-\operatorname{Tr}(Y^H B(x)Y)\right), \qquad Y^\star=B(x)^{-1}S(x).$$2 for single-connected, $$\operatorname{Tr}\!\big(S(x)^H B(x)^{-1}S(x)\big) = \max_Y \left(2\operatorname{Re}\operatorname{Tr}(Y^H S(x))-\operatorname{Tr}(Y^H B(x)Y)\right), \qquad Y^\star=B(x)^{-1}S(x).$$3 for group-connected with $$\operatorname{Tr}\!\big(S(x)^H B(x)^{-1}S(x)\big) = \max_Y \left(2\operatorname{Re}\operatorname{Tr}(Y^H S(x))-\operatorname{Tr}(Y^H B(x)Y)\right), \qquad Y^\star=B(x)^{-1}S(x).$$4, $$\operatorname{Tr}\!\big(S(x)^H B(x)^{-1}S(x)\big) = \max_Y \left(2\operatorname{Re}\operatorname{Tr}(Y^H S(x))-\operatorname{Tr}(Y^H B(x)Y)\right), \qquad Y^\star=B(x)^{-1}S(x).$$5 for group-connected with $$\operatorname{Tr}\!\big(S(x)^H B(x)^{-1}S(x)\big) = \max_Y \left(2\operatorname{Re}\operatorname{Tr}(Y^H S(x))-\operatorname{Tr}(Y^H B(x)Y)\right), \qquad Y^\star=B(x)^{-1}S(x).$$6, and $$\operatorname{Tr}\!\big(S(x)^H B(x)^{-1}S(x)\big) = \max_Y \left(2\operatorname{Re}\operatorname{Tr}(Y^H S(x))-\operatorname{Tr}(Y^H B(x)Y)\right), \qquad Y^\star=B(x)^{-1}S(x).$$7 for fully connected BD-RIS, compared with approximately $$\operatorname{Tr}\!\big(S(x)^H B(x)^{-1}S(x)\big) = \max_Y \left(2\operatorname{Re}\operatorname{Tr}(Y^H S(x))-\operatorname{Tr}(Y^H B(x)Y)\right), \qquad Y^\star=B(x)^{-1}S(x).$$8, $$\operatorname{Tr}\!\big(S(x)^H B(x)^{-1}S(x)\big) = \max_Y \left(2\operatorname{Re}\operatorname{Tr}(Y^H S(x))-\operatorname{Tr}(Y^H B(x)Y)\right), \qquad Y^\star=B(x)^{-1}S(x).$$9, $f(\gamma)=\log(1+\gamma)$0, and $f(\gamma)=\log(1+\gamma)$1 in the cited manifold-only baseline (Fidanovski et al., 10 Nov 2025).

In tri-hybrid beamforming for radiation-center reconfigurable arrays, energy-efficiency maximization introduces a nested fractional structure. The cited DQTFP scheme uses one quadratic transform for the outer energy-efficiency ratio through an auxiliary $f(\gamma)=\log(1+\gamma)$2 and a second quadratic transform for the inner SINR ratios through auxiliaries $f(\gamma)=\log(1+\gamma)$3. This yields a penalized subproblem in the vectorized beamformer that is convex and solved by CVX, while analog and digital beamformers are updated in closed form within a tri-loop alternating-optimization framework. A lower-complexity alternative, termed LDTFP in that paper, replaces the CVX-based inner step with closed-form matrix updates. For the reported FC-HBF setting at input SNR around $f(\gamma)=\log(1+\gamma)$4 dB, LDTFP runs in $f(\gamma)=\log(1+\gamma)$5 ms versus $f(\gamma)=\log(1+\gamma)$6 s for DQTFP, with comparable energy efficiency when SNR is at most $f(\gamma)=\log(1+\gamma)$7 dB (Li et al., 21 Aug 2025).

A later critique of classical LDT+QT appears in joint uplink scheduling and power control for multicell networks. There, the classical closed-form FP pipeline—identified explicitly as LDT+QT—remains attractive because it preserves per-cell separability in a mixed discrete–continuous problem. However, the paper argues that the LDT surrogate is conservative and proposes a reciprocal-inversion transform compatible with QT. The resulting SEFP algorithm retains closed-form auxiliary, scheduling, and power updates while providing a uniformly tighter surrogate than classical LDT+QT. Reported quantitative gains include a proportional-fairness utility of $f(\gamma)=\log(1+\gamma)$8 versus $f(\gamma)=\log(1+\gamma)$9 for classical FP, an equal-weight sum-rate mean of $$\log(1+\gamma) = \max_{u\ge 0} \left[ \log(1+u)-u+(1+u)\frac{\gamma}{1+\gamma} \right], \qquad u^\star=\gamma.$$0 versus $$\log(1+\gamma) = \max_{u\ge 0} \left[ \log(1+u)-u+(1+u)\frac{\gamma}{1+\gamma} \right], \qquad u^\star=\gamma.$$1, and a normalized random-priority WSR mean of $$\log(1+\gamma) = \max_{u\ge 0} \left[ \log(1+u)-u+(1+u)\frac{\gamma}{1+\gamma} \right], \qquad u^\star=\gamma.$$2 versus $$\log(1+\gamma) = \max_{u\ge 0} \left[ \log(1+u)-u+(1+u)\frac{\gamma}{1+\gamma} \right], \qquad u^\star=\gamma.$$3 (Jiao et al., 2 Jul 2026).

These developments clarify a final point often obscured by the acronym. DQTFP is not tied to one solver architecture or one transform sequence. It has been embedded in SDR, LP reformulation, low-rank factorization, manifold optimization, and nested beamforming decompositions; it has also been challenged by tighter surrogate constructions when the classical LDT+QT combination becomes conservative. This suggests that DQTFP is most accurately viewed as a fractional-programming paradigm organized around exact or MM-tight ratio decoupling, closed-form auxiliary updates, and structurally matched primal solvers.