Extragradient-FP for RSMA Optimization

• Extragradient-Fractional Programming (EG-FP) is an optimization method that reformulates the non-convex MMF problem in RSMA via fractional programming for joint beamforming and rate allocation.
• It integrates an extragradient solver with a variational inequality approach to achieve MMF rates comparable to successive convex approximation while using significantly less computational time.
• The low-dimensional variant exploits the beamforming subspace structure to decouple complexity from the number of antennas, making it highly effective for massive MIMO deployments.

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 (Luo et al., 6 Jul 2025). 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 (Luo et al., 6 Jul 2025).

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 (Luo et al., 6 Jul 2025). 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

$N_t$0

For user $N_t$1, the SINRs for the common and private streams are

$N_t$2

The corresponding rates are $N_t$3 and $N_t$4, and the decodable common-stream rate is

$N_t$5

With nonnegative common-rate shares $N_t$6 satisfying $N_t$7, the total rate of user $N_t$8 is $N_t$9.

The MMF design problem is

$K$0

This problem is non-convex because of the SINR ratios and non-smooth because of the nested minimum operators (Luo et al., 6 Jul 2025). Conventional approaches are reported to incur either high computational complexity or degraded MMF performance, which motivates the EG-FP construction (Luo et al., 6 Jul 2025).

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 (Luo et al., 6 Jul 2025). The first step uses the Lagrangian transform to extract the SINR fractions by introducing auxiliary variables $K$1. For the common-stream term,

$K$2

with equality at $K$3. An analogous construction holds for the private-stream term $K$4 (Luo et al., 6 Jul 2025).

A second FP step uses the quadratic transform to decouple the numerator and denominator of each SINR by introducing complex auxiliary variables $K$5. For the common stream,

$K$6

which satisfies $K$7, with equality at

$K$8

Again, the same construction applies to the private-stream terms $K$9 (Luo et al., 6 Jul 2025).

Substituting these transforms yields an equivalent problem in $\mathcal{K}=\{1,\dots,K\}$0:

$\mathcal{K}=\{1,\dots,K\}$1

Although still non-convex globally, this transformed problem is block-wise convex in three blocks: $\mathcal{K}=\{1,\dots,K\}$2, $\mathcal{K}=\{1,\dots,K\}$3, and $\mathcal{K}=\{1,\dots,K\}$4 (Luo et al., 6 Jul 2025). 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 $\mathcal{K}=\{1,\dots,K\}$5 for the MMF objective:

$\mathcal{K}=\{1,\dots,K\}$6

This subproblem is convex (Luo et al., 6 Jul 2025). Its Lagrangian is formed with multipliers $\mathcal{K}=\{1,\dots,K\}$7 and $\mathcal{K}=\{1,\dots,K\}$8:

$\mathcal{K}=\{1,\dots,K\}$9

The corresponding dual problem is

$k$0

and strong duality is reported to hold (Luo et al., 6 Jul 2025).

To reveal the computational structure, the complex beamforming variables are split into real and imaginary parts, all primal variables are stacked into $k$1, all dual variables into $k$2, and

$k$3

The saddle-point KKT condition is then exactly the VI condition

$k$4

where $k$5 is the Cartesian product of Euclidean primal space and the nonnegative orthant for the dual variables (Luo et al., 6 Jul 2025). Because the Lagrangian is concave in the primal variables and convex in the multipliers, the mapping $k$6 is monotone on $k$7 (Luo et al., 6 Jul 2025).

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 (Luo et al., 6 Jul 2025). Given the current iterate $k$8 and a stepsize $k$9, the algorithm performs a prediction step followed by a correction step:

$W_k$0

$W_k$1

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 (Luo et al., 6 Jul 2025).

The two-step structure is described as essential: a one-step gradient ascent/descent procedure would diverge on the saddle-point problem (Luo et al., 6 Jul 2025). Stepsizes are selected according to the rule

$W_k$2

with a practical cap $W_k$3 to avoid excessively large updates (Luo et al., 6 Jul 2025). Under monotonicity of $W_k$4, this rule ensures

$W_k$5

for any solution $W_k$6, hence convergence (Luo et al., 6 Jul 2025).

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 (Luo et al., 6 Jul 2025). The argument is that each $W_k$7 update maximizes the current surrogate objective and each $W_k$8 update exactly attains the FP bound, so the MMF value cannot decrease and is upper bounded by the power constraint (Luo et al., 6 Jul 2025).

For tolerances $W_k$9 and $W_{c,k}$0 for the outer and inner loops respectively, the reported iteration complexity is

$W_{c,k}$1

and

$W_{c,k}$2

These expressions are reported after combining the AO outer-loop complexity with the extragradient inner-loop complexity (Luo et al., 6 Jul 2025).

5. Low-dimensional beamforming structure

A central structural result is Theorem 1: every optimal beamformer $W_{c,k}$3, for $W_{c,k}$4, lies in the column space of the channel matrix $W_{c,k}$5 (Luo et al., 6 Jul 2025). Equivalently,

$W_{c,k}$6

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 (Luo et al., 6 Jul 2025).

Defining

$W_{c,k}$7

the original MMF design can be reduced to a lower-dimensional problem:

$W_{c,k}$8

where $W_{c,k}$9 is the $W_{p,k}$0-th column of $W_{p,k}$1 (Luo et al., 6 Jul 2025).

The computational consequence is explicit. In the full-dimensional AO inner loop, the gradient with respect to $W_{p,k}$2 costs $W_{p,k}$3 per iteration. After reparameterization, the low-dimensional variable $W_{p,k}$4 costs $W_{p,k}$5 per iteration, plus a one-time $W_{p,k}$6 cost to form $W_{p,k}$7 (Luo et al., 6 Jul 2025). Thus, when $W_{p,k}$8, 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

$W_{p,k}$9

(Luo et al., 6 Jul 2025). The conditional-average common and private rates are

$s_c$0

Using Jensen’s inequality and treating $s_c$1 as Gaussian noise yields closed-form lower bounds such as

$s_c$2

with an analogous expression for $s_c$3 (Luo et al., 6 Jul 2025). 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 (Luo et al., 6 Jul 2025).

The numerical evaluation reported for the method uses channels $s_c$4, noise variance $s_c$5, SNR $s_c$6 from $s_c$7 dB to $s_c$8 dB, problem sizes including $s_c$9, $s_1,\dots,s_K$0 realizations, and tolerances $s_1,\dots,s_K$1 (Luo et al., 6 Jul 2025). The algorithms compared are EG-FP in full and low-dimensional forms, SCA implemented as “FP + CVX,” and GPI (Luo et al., 6 Jul 2025).

The reported empirical findings are consistent across the study. EG-FP and low-dimensional EG-FP achieve MMF rates within $s_1,\dots,s_K$2 of SCA across all $s_1,\dots,s_K$3, while outperforming GPI by up to $s_1,\dots,s_K$4–$s_1,\dots,s_K$5 (Luo et al., 6 Jul 2025). In runtime, EG-FP uses $s_1,\dots,s_K$6 of SCA’s runtime and is $s_1,\dots,s_K$7–$s_1,\dots,s_K$8 faster than GPI for moderate $s_1,\dots,s_K$9; the low-dimensional version delivers further speedups as $N_t$00, including a reported $N_t$01-antenna case (Luo et al., 6 Jul 2025). 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 (Luo et al., 6 Jul 2025).

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 (Luo et al., 6 Jul 2025). 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 (Luo et al., 6 Jul 2025).