---
title: Extragradient-FP for RSMA Optimization
url: https://www.emergentmind.com/topics/extragradient-fractional-programming-eg-fp
type: topic
---

# Extragradient-FP for RSMA Optimization

Searching arXiv for the cited paper and closely related references.
Searching arXiv for “An Efficient Max-Min Fair Resource Optimization Algorithm for Rate-Splitting Multiple Access”.
Extragradient-Fractional Programming (EG-FP) is an optimization algorithm proposed for the max-min fairness (MMF) problem in downlink rate-splitting multiple access (RSMA), where the joint design of beamforming and common-rate allocation is non-convex, non-smooth, and strongly coupled [2507.04201]. The method combines fractional programming (FP), which reformulates the original SINR-constrained MMF objective into a block-wise convex form, with an extragradient solver applied to a variational inequality (VI) representation of the resulting precoding subproblem. In the formulation reported by Luo and Mao, EG-FP yields MMF rates that closely match those of a conventional successive convex approximation (SCA) algorithm while requiring less than \(10\%\) of the average CPU time of SCA; a low-dimensional variant further removes dependence of computational complexity on the number of transmit antennas, which is especially beneficial when the base station employs a large antenna array [2507.04201].

## 1. Problem setting in downlink RSMA

The algorithm is developed for a downlink MU-MISO system with an \(N_t\)-antenna base station serving \(K\) single-antenna users \(\mathcal{K}=\{1,\dots,K\}\) using 1-layer rate splitting [2507.04201]. User \(k\)'s message \(W_k\) is divided into a common part \(W_{c,k}\) and a private part \(W_{p,k}\). The common parts are jointly encoded into a common stream \(s_c\), while the private parts are encoded into user-specific streams \(s_1,\dots,s_K\). The transmit signal is
$$
\mathbf{x}=\mathbf{p}_c s_c+\sum_{k=1}^K \mathbf{p}_k s_k,
\qquad
\mathrm{tr}(\mathbf{P}^H\mathbf{P})\le P_t,
\qquad
\mathbf{P}=[\mathbf{p}_c,\mathbf{p}_1,\dots,\mathbf{p}_K]\in\mathbb{C}^{N_t\times(K+1)}.
$$

For user \(k\), the SINRs for the common and private streams are
$$
\gamma_{c,k}=
\frac{|\mathbf{h}_k^H\mathbf{p}_c|^2}
{\sum_{j=1}^K|\mathbf{h}_k^H\mathbf{p}_j|^2+\sigma_k^2},
\qquad
\gamma_{p,k}=
\frac{|\mathbf{h}_k^H\mathbf{p}_k|^2}
{\sum_{j\neq k}|\mathbf{h}_k^H\mathbf{p}_j|^2+\sigma_k^2}.
$$
The corresponding rates are \(R_{c,k}=\ln(1+\gamma_{c,k})\) and \(R_{p,k}=\ln(1+\gamma_{p,k})\), and the decodable common-stream rate is
$$
R_c=\min_{k\in\mathcal{K}} R_{c,k}.
$$
With nonnegative common-rate shares \(c_k\) satisfying \(\sum_k c_k\le R_c\), the total rate of user \(k\) is \(R_{p,k}+c_k\).

The MMF design problem is
$$
\begin{aligned}
\max_{\mathbf{P},\,\mathbf{c}\ge 0}
&\;\Bigl\{\min_{k\in\mathcal{K}} (R_{p,k}+c_k)\Bigr\} \\
\text{s.t.}\quad
&\sum_{k=1}^K c_k\le R_c, \\
&\mathrm{tr}(\mathbf{P}^H\mathbf{P})\le P_t.
\end{aligned}
$$
This problem is non-convex because of the SINR ratios and non-smooth because of the nested minimum operators [2507.04201]. Conventional approaches are reported to incur either high computational complexity or degraded MMF performance, which motivates the EG-FP construction [2507.04201].

## 2. Fractional-programming transformation

EG-FP begins by applying FP to transform the original MMF problem into an equivalent formulation that is block-wise convex [2507.04201]. The first step uses the Lagrangian transform to extract the SINR fractions by introducing auxiliary variables \(\vartheta_{c,k},\vartheta_{p,k}\ge 0\). For the common-stream term,
$$
f_{c,k}(\mathbf{P},\vartheta_{c,k})
=
\ln(1+\vartheta_{c,k})
-\vartheta_{c,k}
+\frac{(1+\vartheta_{c,k})\gamma_{c,k}}{1+\gamma_{c,k}}
\le R_{c,k},
$$
with equality at \(\vartheta_{c,k}^\star=\gamma_{c,k}\). An analogous construction holds for the private-stream term \(f_{p,k}\) [2507.04201].

A second FP step uses the quadratic transform to decouple the numerator and denominator of each SINR by introducing complex auxiliary variables \(\varphi_{c,k},\varphi_{p,k}\). For the common stream,
$$
g_{c,k}(\mathbf{P},\vartheta_{c,k},\varphi_{c,k})
=
\ln(1+\vartheta_{c,k})-\vartheta_{c,k}
+2\Re\{\varphi_{c,k}^H \mathbf{h}_k^H\mathbf{p}_c\}
-|\varphi_{c,k}|^2\Bigl(\sum_j|\mathbf{h}_k^H\mathbf{p}_j|^2+\sigma_k^2\Bigr),
$$
which satisfies \(g_{c,k}\le f_{c,k}\le R_{c,k}\), with equality at
$$
\varphi_{c,k}^\star
=
\frac{\sqrt{1+\vartheta_{c,k}}\;\mathbf{h}_k^H\mathbf{p}_c}
{\sum_j|\mathbf{h}_k^H\mathbf{p}_j|^2+\sigma_k^2}.
$$
Again, the same construction applies to the private-stream terms \(g_{p,k}\) [2507.04201].

Substituting these transforms yields an equivalent problem in \((\mathbf{P},\mathbf{c},\bm{\vartheta},\bm{\varphi})\):
$$
\begin{aligned}
\max_{\substack{\mathbf{P},\mathbf{c}\ge0,\\
\bm\vartheta_c,\bm\vartheta_p\ge0,\\
\bm\varphi_c,\bm\varphi_p}}
&\;\min_k\{g_{p,k}(\mathbf{P},\vartheta_{p,k},\varphi_{p,k})+c_k\} \\
\text{s.t.}\quad
&\sum_k c_k\le \min_k g_{c,k}(\mathbf{P},\vartheta_{c,k},\varphi_{c,k}),\\
&\mathrm{tr}(\mathbf{P}^H\mathbf{P})\le P_t.
\end{aligned}
$$
Although still non-convex globally, this transformed problem is block-wise convex in three blocks: \((\mathbf{P},\mathbf{c})\), \((\bm\vartheta_c,\bm\vartheta_p)\), and \((\bm\varphi_c,\bm\varphi_p)\) [2507.04201]. The last two blocks admit closed-form updates via stationary conditions, while the precoding-and-rate block is handled by the extragradient stage.

This suggests that EG-FP should be understood not as a stand-alone first-order scheme, but as a hybrid FP/AO procedure in which FP isolates convex substructure and the extragradient method resolves the remaining saddle-point geometry of the core subproblem.

## 3. Dual reformulation and variational inequality structure

When the FP auxiliary variables are fixed, the remaining subproblem can be written by introducing an auxiliary scalar \(t\) for the MMF objective:
$$
\begin{aligned}
\max_{\mathbf{P},\mathbf{c}\ge0,\,t}\quad
&t\\
\text{s.t.}\quad
&t\le c_k+g_{p,k}(\mathbf{P}),\quad \forall k,\\
&\sum_j c_j\le g_{c,k}(\mathbf{P}),\quad \forall k,\\
&\mathrm{tr}(\mathbf{P}^H\mathbf{P})\le P_t.
\end{aligned}
$$
This subproblem is convex [2507.04201]. Its Lagrangian is formed with multipliers \(\bm\lambda,\bm\rho,\bm\mu\ge 0\) and \(\omega\ge 0\):
$$
\begin{aligned}
L(\mathbf{P},\mathbf{c},t;\bm\lambda,\bm\rho,\bm\mu,\omega)
&=
t-\sum_k \lambda_k\bigl(t-(c_k+g_{p,k}(\mathbf{P}))\bigr)\\
&\quad
-\sum_k \rho_k\bigl(\sum_j c_j-g_{c,k}(\mathbf{P})\bigr)
+\sum_k \mu_k c_k
-\omega\bigl(\mathrm{tr}(\mathbf{P}^H\mathbf{P})-P_t\bigr).
\end{aligned}
$$
The corresponding dual problem is
$$
\min_{\bm\lambda,\bm\rho,\bm\mu\ge0,\;\omega\ge0}\;
\max_{\mathbf{P},\mathbf{c},t}\;
L(\mathbf{P},\mathbf{c},t;\bm\lambda,\bm\rho,\bm\mu,\omega),
$$
and strong duality is reported to hold [2507.04201].

To reveal the computational structure, the complex beamforming variables are split into real and imaginary parts, all primal variables are stacked into \(\mathbf{y}\), all dual variables into \(\mathbf{z}\), and
$$
\mathbf{x}=[\mathbf{y}^T,\mathbf{z}^T]^T,
\qquad
\mathbf{h}(\mathbf{x})
=
\begin{bmatrix}
-\nabla_{\mathbf{y}}L\\[2pt]
\nabla_{\mathbf{z}}L
\end{bmatrix}.
$$
The saddle-point KKT condition is then exactly the VI condition
$$
\mathbf{h}(\mathbf{x}^\star)^T(\mathbf{x}-\mathbf{x}^\star)\ge 0,
\qquad
\forall\,\mathbf{x}\in\mathcal{S},
$$
where \(\mathcal{S}\) is the Cartesian product of Euclidean primal space and the nonnegative orthant for the dual variables [2507.04201]. Because the Lagrangian is concave in the primal variables and convex in the multipliers, the mapping \(\mathbf{h}(\cdot)\) is monotone on \(\mathcal{S}\) [2507.04201].

The significance of this reformulation is methodological. Rather than solving the convex subproblem by generic conic or interior-point machinery, EG-FP exploits the monotone VI form directly. A plausible implication is that the algorithm’s empirical speed advantage over SCA is tied not only to FP itself, but also to replacing toolbox-based convex optimization with a structure-aware first-order saddle-point solver.

## 4. Extragradient iterations and convergence properties

The VI is solved with Korpelevich’s extragradient method [2507.04201]. Given the current iterate \(\mathbf{x}^{(n)}\in\mathcal{S}\) and a stepsize \(\alpha^{(n)}>0\), the algorithm performs a prediction step followed by a correction step:
$$
\bar{\mathbf{x}}^{(n)}
=
\operatorname{Proj}_{\mathcal{S}}
\bigl[
\mathbf{x}^{(n)}-\alpha^{(n)}\mathbf{h}(\mathbf{x}^{(n)})
\bigr],
$$
$$
\mathbf{x}^{(n+1)}
=
\operatorname{Proj}_{\mathcal{S}}
\bigl[
\mathbf{x}^{(n)}-\alpha^{(n)}\mathbf{h}(\bar{\mathbf{x}}^{(n)})
\bigr].
$$
The projection enforces nonnegativity of dual variables, while the primal variables remain unconstrained in the projected set because the power and rate restrictions are absorbed in the Lagrangian representation [2507.04201].

The two-step structure is described as essential: a one-step gradient ascent/descent procedure would diverge on the saddle-point problem [2507.04201]. Stepsizes are selected according to the rule
$$
\alpha^{(n)}
=
\beta
\frac{\|\mathbf{x}^{(n)}-\bar{\mathbf{x}}^{(n)}\|}
{\|\mathbf{h}(\mathbf{x}^{(n)})-\mathbf{h}(\bar{\mathbf{x}}^{(n)})\|},
\qquad
\beta\in(0,1),
$$
with a practical cap \(\alpha^{(n)}\le \alpha_{\max}\) to avoid excessively large updates [2507.04201]. Under monotonicity of \(\mathbf{h}\), this rule ensures
\[
\|\mathbf{x}^{(n+1)}-\mathbf{x}^\star\|<\|\mathbf{x}^{(n)}-\mathbf{x}^\star\|
\]
for any solution \(\mathbf{x}^\star\), hence convergence [2507.04201].

Within the full EG-FP procedure, the extragradient iterations form the inner loop, while the FP auxiliary variables are updated in closed form in an outer alternating-optimization (AO) loop. The reported convergence result states that if the inner loop converges to the optimum of the convex subproblem, then the outer FP/AO iterations generate a non-decreasing MMF objective and converge [2507.04201]. The argument is that each \((\mathbf{P},\mathbf{c})\) update maximizes the current surrogate objective and each \((\bm\vartheta,\bm\varphi)\) update exactly attains the FP bound, so the MMF value cannot decrease and is upper bounded by the power constraint [2507.04201].

For tolerances \(\epsilon_1\) and \(\epsilon_2\) for the outer and inner loops respectively, the reported iteration complexity is
$$
O\!\Bigl(\log\!\bigl(\tfrac1{\epsilon_1}\bigr)\sqrt{\tfrac1{\epsilon_2}}\,K^2N_t\Bigr)
\quad\text{for the full-dimensional algorithm,}
$$
and
$$
O\!\Bigl(\log\!\bigl(\tfrac1{\epsilon_1}\bigr)\sqrt{\tfrac1{\epsilon_2}}\,K^3\Bigr)
\quad\text{for the low-dimensional variant.}
$$
These expressions are reported after combining the AO outer-loop complexity with the extragradient inner-loop complexity [2507.04201].

## 5. Low-dimensional beamforming structure

A central structural result is Theorem 1: every optimal beamformer \(\mathbf{p}_k\), for \(k\in\{c,1,\dots,K\}\), lies in the column space of the channel matrix \(\mathbf{H}=[\mathbf{h}_1,\dots,\mathbf{h}_K]\) [2507.04201]. Equivalently,
$$
\mathbf{p}_k=\mathbf{H}\mathbf{q}_k,
\qquad
\mathbf{q}_k\in\mathbb{C}^{K}.
$$
The proof sketch reported in the source is based on KKT stationarity of the convex subproblem, which shows that each beamformer is a linear combination of the user channels [2507.04201].

Defining
\[
\mathbf{Q}=[\mathbf{q}_c,\mathbf{q}_1,\dots,\mathbf{q}_K]\in\mathbb{C}^{K\times(K+1)},
\qquad
\mathbf{G}=\mathbf{H}^H\mathbf{H},
\]
the original MMF design can be reduced to a lower-dimensional problem:
$$
\begin{aligned}
\max_{\mathbf{Q},\mathbf{c}\ge0}
&\;\min_k\Bigl\{
\ln\!\Bigl(1+\frac{|\mathbf{g}_k^H\mathbf{q}_k|^2}
{\sum_{j\neq k}|\mathbf{g}_k^H\mathbf{q}_j|^2+\sigma_k^2}\Bigr)
+c_k
\Bigr\}\\
\text{s.t.}\quad
&\sum_k c_k\le
\min_k
\ln\!\Bigl(
1+\frac{|\mathbf{g}_k^H\mathbf{q}_c|^2}
{\sum_j|\mathbf{g}_k^H\mathbf{q}_j|^2+\sigma_k^2}
\Bigr),\\
&\mathrm{tr}(\mathbf{Q}^H\mathbf{G}\mathbf{Q})\le P_t,
\end{aligned}
$$
where \(\mathbf{g}_k\) is the \(k\)-th column of \(\mathbf{G}\) [2507.04201].

The computational consequence is explicit. In the full-dimensional AO inner loop, the gradient with respect to \(\mathbf{P}\) costs \(\mathcal{O}(K^2N_t)\) per iteration. After reparameterization, the low-dimensional variable \(\mathbf{Q}\) costs \(\mathcal{O}(K^3)\) per iteration, plus a one-time \(\mathcal{O}(K^2N_t)\) cost to form \(\mathbf{G}\) [2507.04201]. Thus, when \(N_t\gg K\), the per-iteration complexity becomes independent of the number of transmit antennas.

This low-dimensional structure is one of the most distinctive features of EG-FP. It is not merely an implementation trick; it changes the scaling law of the optimization routine. A plausible implication is that the low-dimensional version is especially relevant for massive-MIMO-like deployments in which antenna growth would otherwise dominate the cost of iterative beamforming updates.

## 6. Extension to imperfect CSIT and empirical behavior

The method is extended to imperfect CSIT by modeling the true channel as
\[
\mathbf{h}_k=\widehat{\mathbf{h}}_k+\mathbf{e}_k,
\qquad
\mathbb{E}[\mathbf{h}_k\mid \widehat{\mathbf{h}}_k]=\widehat{\mathbf{h}}_k,
\qquad
\mathbb{E}[\mathbf{e}_k\mathbf{e}_k^H]=\mathbf{R}_k
\]
[2507.04201]. The conditional-average common and private rates are
$$
\widehat{R}_{c,k}(\widehat{\mathbf{H}})
=
\mathbb{E}_{\mathbf{H}\mid \widehat{\mathbf{H}}}
\bigl[\ln(1+\gamma_{c,k})\bigr],
\qquad
\widehat{R}_{p,k}(\widehat{\mathbf{H}})
=
\mathbb{E}_{\mathbf{H}\mid \widehat{\mathbf{H}}}
\bigl[\ln(1+\gamma_{p,k})\bigr].
$$
Using Jensen’s inequality and treating \(\mathbf{e}_k^H\mathbf{P}s\) as Gaussian noise yields closed-form lower bounds such as
$$
\widehat{R}_{c,k}^{\mathrm{lb}}
=
\ln\!\Bigl(
1+
\frac{|\widehat{\mathbf{h}}_k^H\mathbf{p}_c|^2}
{\sum_j|\widehat{\mathbf{h}}_k^H\mathbf{p}_j|^2+\sum_j \mathbf{p}_j^H\mathbf{R}_k\mathbf{p}_j+\sigma_k^2}
\Bigr),
$$
with an analogous expression for \(\widehat{R}_{p,k}^{\mathrm{lb}}\) [2507.04201]. Replacing the perfect-CSIT rates in the MMF problem with these lower bounds produces a deterministic lower-bound problem with the same structure, so EG-FP and its low-dimensional variant apply without change [2507.04201].

The numerical evaluation reported for the method uses channels \(\mathbf{h}_k\sim\mathcal{CN}(0,\mathbf{I})\), noise variance \(\sigma_k^2=1\), SNR \(P_t\) from \(0\) dB to \(20\) dB, problem sizes including \((K,N_t)=(4,16),(16,64),\dots\), \(100\) realizations, and tolerances \(\epsilon_1=\epsilon_2=10^{-3}\) [2507.04201]. The algorithms compared are EG-FP in full and low-dimensional forms, SCA implemented as “FP + CVX,” and GPI [2507.04201].

The reported empirical findings are consistent across the study. EG-FP and low-dimensional EG-FP achieve MMF rates within \(<1\%\) of SCA across all \((K,N_t)\), while outperforming GPI by up to \(10\)–\(20\%\) [2507.04201]. In runtime, EG-FP uses \(<10\%\) of SCA’s runtime and is \(2\times\)–\(5\times\) faster than GPI for moderate \(K,N_t\); the low-dimensional version delivers further speedups as \(N_t\gg K\), including a reported \(256\)-antenna case [2507.04201]. Under imperfect CSIT, the ergodic MMF rates of EG-FP are reported to closely track the SCA lower bound while running an order of magnitude faster, whereas GPI suffers a severe performance drop under CSIT error [2507.04201].

A common misconception would be to interpret EG-FP as sacrificing fairness performance for speed. The reported results do not support that interpretation: the stated behavior is that MMF performance closely matches SCA while computational cost is substantially lower [2507.04201]. Another possible misconception is that its efficiency depends only on the first-order solver. The paper attributes efficiency to the combined effect of FP reformulation, VI-based extragradient updates, and the beamforming subspace reduction [2507.04201].

Source: https://www.emergentmind.com/topics/extragradient-fractional-programming-eg-fp