Sequential Rank-One Constraint Relaxation (SROCR)

• SROCR is an iterative process that replaces rank-one constraints with sequential convex inequalities, ensuring the dominant eigenvalue approximates the full trace.
• The technique leverages the lifted semidefinite formulation to progressively tighten spectral conditions using previous iterates and anchored eigenvectors.
• Applications in waveform design, robust ISAC, and RIS-aided networks demonstrate that SROCR enhances rank recovery and performance compared to conventional SDR approaches.

Searching arXiv for SROCR-related papers and the target paper to ground the encyclopedia entry. First, I’ll retrieve the main target paper and several additional papers that explicitly use SROCR in different optimization settings. Using the arXiv search tool now. Sequential Rank-One Constraint Relaxation (SROCR) is an iterative methodology for optimization problems in which semidefinite lifting introduces positive semidefinite matrix variables that are required to be rank-one. Rather than discarding the rank constraint outright, SROCR replaces it with a sequence of convex surrogate inequalities anchored at the dominant eigendirection of the current iterate, and progressively tightens these inequalities until the largest eigenvalue accounts for essentially the entire trace. In recent arXiv literature, this mechanism is used in continuous-Doppler unimodular waveform design, robust near-field integrated sensing and communication (ISAC), STAR-RIS- and ARIS-assisted ISAC, near-field STAR-RIS-enabled integrated sensing, communication, and power transfer (ISCPT), and RIS-aided fluid-antenna UAV networks (Lin et al., 8 Apr 2025, Chen et al., 17 Jul 2025, Liu et al., 2024, Jin et al., 6 Feb 2026, Rostamikafaki et al., 15 May 2026, Shen et al., 16 Jan 2025).

1. Problem class and lifted rank-one structure

SROCR is applied after a nonconvex vector design has been lifted into a matrix variable. In the continuous-Doppler waveform problem, a unimodular sequence $\boldsymbol{x}$ is lifted to $$X=\boldsymbol{x}\boldsymbol{x}^H \in \mathbb{H}_+^N$$ with $\operatorname{rank}(X)=1$, while unimodularity is enforced by diagonal equalities $\operatorname{Tr}(E^{(n)}X)=1$. In robust beamforming, the same pattern appears as $W_k=w_k w_k^H \succeq 0$ with $\operatorname{rank}(W_k)=1$. STAR-RIS and ARIS formulations further introduce lifted passive variables such as $V_t=\nu_t\nu_t^H$, $V_r=\nu_r\nu_r^H$, or $\bar{\Phi}=\bar{\phi}\bar{\phi}^H$, again with rank-one requirements. In RIS-aided fluid-antenna UAV networks, both the active transmit covariances $F_k=w_k w_k^H$ and the passive RIS covariance $$X=\boldsymbol{x}\boldsymbol{x}^H \in \mathbb{H}_+^N$$0 are handled in this lifted form (Lin et al., 8 Apr 2025, Chen et al., 17 Jul 2025, Liu et al., 2024, Jin et al., 6 Feb 2026, Shen et al., 16 Jan 2025).

The rank-one constraint is the point at which the convexity of the lifted problem is lost. The surrounding constraints—power budgets, linear matrix inequalities (LMIs), diagonal constraints, or first-order convex surrogates—are typically convex once the lifting has been performed. SROCR therefore acts specifically on the residual nonconvexity created by the requirement that the lifted covariance correspond to a single physical beamformer, waveform, or passive coefficient vector.

A notable variation appears in near-field STAR-RIS-enabled ISCPT, where SROCR is used for the active covariances $$X=\boldsymbol{x}\boldsymbol{x}^H \in \mathbb{H}_+^N$$1, while the passive STAR-RIS covariances $$X=\boldsymbol{x}\boldsymbol{x}^H \in \mathbb{H}_+^N$$2 are instead treated by a penalty-based successive convex approximation (SCA) rank surrogate of the form $$X=\boldsymbol{x}\boldsymbol{x}^H \in \mathbb{H}_+^N$$3 (Rostamikafaki et al., 15 May 2026). This separation highlights that SROCR is not tied to any one class of matrix variable; it is selected when a sequential spectral tightening is preferable to penalty-based rank control.

2. Spectral basis of the relaxation

The basic identity underlying SROCR is the rank-one characterization for positive semidefinite matrices:

$$X=\boldsymbol{x}\boldsymbol{x}^H \in \mathbb{H}_+^N$$4

In the waveform formulation, this is paired with the Rayleigh–Ritz expression

$$X=\boldsymbol{x}\boldsymbol{x}^H \in \mathbb{H}_+^N$$5

where $$X=\boldsymbol{x}\boldsymbol{x}^H \in \mathbb{H}_+^N$$6 is a unit-norm principal eigenvector. SROCR then replaces the nonconvex equality by an anchored inequality based on the previous iterate:

$$X=\boldsymbol{x}\boldsymbol{x}^H \in \mathbb{H}_+^N$$7

When $$X=\boldsymbol{x}\boldsymbol{x}^H \in \mathbb{H}_+^N$$8, as in the unimodular waveform problem, this becomes

$$X=\boldsymbol{x}\boldsymbol{x}^H \in \mathbb{H}_+^N$$9

As $\operatorname{rank}(X)=1$0, the feasible set shrinks toward $\operatorname{rank}(X)=1$1, i.e. toward rank-one (Lin et al., 8 Apr 2025).

Beamforming papers instantiate the same idea with different notation. The robust near-field ISAC formulation uses

$\operatorname{rank}(X)=1$2

the STAR-RIS-enabled ISAC formulation uses analogous constraints for $\operatorname{rank}(X)=1$3, $\operatorname{rank}(X)=1$4, $\operatorname{rank}(X)=1$5, and $\operatorname{rank}(X)=1$6, the ARIS-RSMA design uses

$\operatorname{rank}(X)=1$7

and the RIS-aided UAV design imposes

$\operatorname{rank}(X)=1$8

The common mechanism is that the dominant eigenvector is fixed from the previous iterate, so the new constraint is linear in the optimization variable and the subproblem remains convex (Chen et al., 17 Jul 2025, Liu et al., 2024, Jin et al., 6 Feb 2026, Shen et al., 16 Jan 2025, Rostamikafaki et al., 15 May 2026).

This spectral interpretation also explains why SROCR is sharper than a one-shot relaxation. Plain semidefinite relaxation (SDR) keeps only $\operatorname{rank}(X)=1$9 or $\operatorname{Tr}(E^{(n)}X)=1$0 and may return higher-rank matrices. SROCR instead enforces progressive concentration of the trace onto a single eigenspace, thereby preserving a direct path back to a physical vector design.

3. Sequential algorithm and recovery of vector solutions

A standard SROCR workflow begins by solving the rank-relaxed convex problem. In the continuous-Doppler waveform design, the initialization solves the SDP without the SROCR inequality to obtain $\operatorname{Tr}(E^{(n)}X)=1$1, then sets

$\operatorname{Tr}(E^{(n)}X)=1$2

where $\operatorname{Tr}(E^{(n)}X)=1$3 controls the aggressiveness of tightening. Each subsequent iteration solves a convex SDP with the anchored inequality, updates the objective $\operatorname{Tr}(E^{(n)}X)=1$4, computes the new dominant eigenpair, and increases the relaxation parameter. If infeasibility occurs, the safeguard is to reduce the increment by halving the auxiliary step parameter and re-solve. Termination is declared when the tightening variable reaches a threshold close to one, such as $\operatorname{Tr}(E^{(n)}X)=1$5, and the objective stabilizes, e.g.

$\operatorname{Tr}(E^{(n)}X)=1$6

The unimodular sequence is then recovered from the principal eigenpair,

$\operatorname{Tr}(E^{(n)}X)=1$7

with $\operatorname{Tr}(E^{(n)}X)=1$8 ensured by the diagonal trace constraints (Lin et al., 8 Apr 2025).

The robust near-field ISAC letter follows the same pattern at the level of user covariances $\operatorname{Tr}(E^{(n)}X)=1$9. It first solves the rank-relaxed problem, computes dominant eigenvectors, sets $W_k=w_k w_k^H \succeq 0$0 and step sizes such as $W_k=w_k w_k^H \succeq 0$1, and then repeatedly solves the convex subproblem with the current eigenvalue-to-trace lower bound. If the subproblem is feasible, the solution is accepted and the relaxation level is updated as

$W_k=w_k w_k^H \succeq 0$2

If infeasible, the step size is shrunk by a factor of two. The stopping rule is $W_k=w_k w_k^H \succeq 0$3 for all $W_k=w_k w_k^H \succeq 0$4 together with objective stabilization, and the reported convergence is about five iterations under perfect CSI (Chen et al., 17 Jul 2025).

The STAR-RIS-enabled ISAC and ARIS-RSMA formulations embed the same loop inside larger alternating procedures. They use tolerance pairs such as $W_k=w_k w_k^H \succeq 0$5 for rank-tightness and $W_k=w_k w_k^H \succeq 0$6 for objective change, update the tightening variables $W_k=w_k w_k^H \succeq 0$7 or $W_k=w_k w_k^H \succeq 0$8 toward one, and retain the previous iterate when infeasibility is encountered, again with a halving rule for the step parameter $W_k=w_k w_k^H \succeq 0$9 (Liu et al., 2024, Jin et al., 6 Feb 2026). In RIS-aided UAV networks, the dominant eigenvector of the current covariance is recomputed after each SDP solve, and the current ratio $\operatorname{rank}(W_k)=1$0 is reused as the next tightening level (Shen et al., 16 Jan 2025).

Taken together, these implementations show SROCR as a deterministic rank-recovery process: solve a convex relaxation, project the next iterate toward the dominant eigendirection, tighten the required concentration, and extract the physical vector from the principal eigenpair once the matrix is practically rank-one.

4. Integration with SDP reformulation, uncertainty handling, and fractional programming

SROCR is ordinarily not the first reformulation step. In the continuous-Doppler waveform problem, the original design is a semi-infinite program because the ambiguity-function sidelobe constraint must hold for all Doppler shifts in a continuous interval. The paper converts the infinite family of constraints $\operatorname{rank}(W_k)=1$1 for $\operatorname{rank}(W_k)=1$2 into finitely many LMIs using positive trigonometric polynomial theory based on Dumitrescu’s results. The resulting SDP is exact with respect to the interval model: the continuous Doppler constraints are not discretized or approximated, and SROCR enters only after this exact SIP-to-SDP reformulation has produced a finite SDP with a single rank-one variable $\operatorname{rank}(W_k)=1$3 (Lin et al., 8 Apr 2025).

In robust near-field ISAC and near-field STAR-RIS-enabled ISCPT, the precursor step is the S-Procedure. Both papers begin with semi-infinite worst-case constraints under norm-bounded channel uncertainty, then convert them into LMIs with nonnegative multipliers. In the near-field secure ISAC letter, the sensing, user-SINR, and secrecy constraints become block LMIs involving $\operatorname{rank}(W_k)=1$4, $\operatorname{rank}(W_k)=1$5, and $\operatorname{rank}(W_k)=1$6, and the paper states that under norm-bounded convex uncertainty sets with non-empty interior the S-Procedure transformations are equivalent. The ISCPT formulation similarly vectorizes the uncertain cascaded channels and obtains block LMIs for energy harvesting, user-rate, eavesdropper, and target-beampattern constraints before applying SROCR to the active covariances (Chen et al., 17 Jul 2025, Rostamikafaki et al., 15 May 2026).

Other works combine SROCR with fractional and block-coordinate procedures. The STAR-RIS-enabled ISAC paper uses Dinkelbach’s transform for the sensing-SINR fraction, majorization-minimization for quartic STAR-RIS terms, and SDR followed by SROCR for both active and passive lifted matrices. The ARIS-RSMA paper decomposes the problem into receive beamforming, transmit beamforming plus rate splitting, and ARIS reflection updates, using generalized Rayleigh quotients, SCA for the RSMA rate constraints, MM for quartic ARIS terms, and SROCR in the transmit and ARIS blocks. The RIS-aided fluid-antenna UAV paper integrates SROCR into a four-block alternative-optimization procedure together with SCA for UAV deployment and FA position adjustment (Liu et al., 2024, Jin et al., 6 Feb 2026, Shen et al., 16 Jan 2025).

This pattern suggests a precise methodological role: SROCR is not a generic substitute for convexification, but a rank-enforcement layer inserted after the original physics and uncertainty sets have already been converted into convex SDP-compatible constraints.

5. Representative applications and reported performance

The continuous-Doppler waveform paper provides the most direct demonstration of SROCR on a single lifted variable. For $\operatorname{rank}(W_k)=1$7, $\operatorname{rank}(W_k)=1$8, and $\operatorname{rank}(W_k)=1$9, the proposed SROCR-SDP design attains $V_t=\nu_t\nu_t^H$0 dB, versus $V_t=\nu_t\nu_t^H$1 dB for GMBI-$V_t=\nu_t\nu_t^H$2 and $V_t=\nu_t\nu_t^H$3 dB for ISQO. For $V_t=\nu_t\nu_t^H$4, $V_t=\nu_t\nu_t^H$5, and $V_t=\nu_t\nu_t^H$6, the reported values are $V_t=\nu_t\nu_t^H$7 dB, $V_t=\nu_t\nu_t^H$8 dB, and $V_t=\nu_t\nu_t^H$9 dB, respectively. For $V_r=\nu_r\nu_r^H$0, $V_r=\nu_r\nu_r^H$1, and $V_r=\nu_r\nu_r^H$2, the values are $V_r=\nu_r\nu_r^H$3 dB, $V_r=\nu_r\nu_r^H$4 dB, and $V_r=\nu_r\nu_r^H$5 dB. The paper further reports that grid-based designs can generate out-of-grid sidelobes and false alarms at fractional target speeds such as $V_r=\nu_r\nu_r^H$6 m/s when the velocity resolution is about $V_r=\nu_r\nu_r^H$7 m/s, whereas the continuous-Doppler SROCR-based design shows no false alarms in the illustrated range–velocity plots (Lin et al., 8 Apr 2025).

In robust near-field secure ISAC, SROCR is reported to outperform SDR across transmit powers $V_r=\nu_r\nu_r^H$8 and antenna counts $V_r=\nu_r\nu_r^H$9 in terms of minimum sensing beampattern gain. Under the thresholds $\bar{\Phi}=\bar{\phi}\bar{\phi}^H$0 dB for communication users and $\bar{\Phi}=\bar{\phi}\bar{\phi}^H$1 dB for eavesdroppers, the design satisfies the worst-case CU SINR constraints while suppressing the eavesdropper SINR more effectively than SDR. The reported convergence is about five iterations under perfect CSI, and robustness is maintained for normalized CSI errors $\bar{\Phi}=\bar{\phi}\bar{\phi}^H$2 up to $\bar{\Phi}=\bar{\phi}\bar{\phi}^H$3–$\bar{\Phi}=\bar{\phi}\bar{\phi}^H$4 (Chen et al., 17 Jul 2025).

The STAR-RIS-enabled ISAC work reports convergence within few iterations across different BS power budgets and rate thresholds. It also reports that $\bar{\Phi}=\bar{\phi}\bar{\phi}^H$5 shrinks toward zero as the communication rate threshold increases, with $\bar{\Phi}=\bar{\phi}\bar{\phi}^H$6 at $\bar{\Phi}=\bar{\phi}\bar{\phi}^H$7, which corroborates its theorem that sensing-only power is unnecessary in the optimized design. The ARIS-RSMA letter reports that at $\bar{\Phi}=\bar{\phi}\bar{\phi}^H$8 bits/Hz the worst-target echo SINR is $\bar{\Phi}=\bar{\phi}\bar{\phi}^H$9 dB for ARIS-RSMA, versus $F_k=w_k w_k^H$0 dB for ARIS-NOMA and $F_k=w_k w_k^H$1 dB for ARIS-SDMA, with the paper attributing the improvement to the joint MM and SROCR design (Liu et al., 2024, Jin et al., 6 Feb 2026).

In near-field STAR-RIS-enabled ISCPT, the alternating-optimization routine is reported to converge within about six outer iterations, and the SROCR-based active design achieves the highest harvested power among the tested baselines while meeting secrecy and beampattern constraints. In RIS-aided fluid-antenna UAV networks, the BRAUD algorithm with SROCR achieves approximately $F_k=w_k w_k^H$2 gain over “without FA,” approximately $F_k=w_k w_k^H$3 over “without UAV deployment,” approximately $F_k=w_k w_k^H$4 over “without RIS deployment,” and approximately $F_k=w_k w_k^H$5 over “without proper beamforming,” while also outperforming drop-rank, heuristic, zero-forcing, and random benchmarks (Rostamikafaki et al., 15 May 2026, Shen et al., 16 Jan 2025).

6. Relation to SDR, theoretical status, and limitations

A recurring comparison in the literature is between SROCR and plain SDR. SDR simply drops the rank-one constraints and solves the convex SDP; several papers state that this can yield higher-rank matrices that require randomization, eigen-decomposition, or other postprocessing, with possible performance loss or violation of the original structure. By contrast, SROCR keeps the original objective and convex constraints intact and progressively strengthens a spectral concentration condition. The waveform paper explicitly contrasts SROCR with SDR, reweighted nuclear norm or log-det penalties, and ADMM-based rank minimization, emphasizing that SROCR does not modify the objective or inject penalty terms. The STAR-RIS-enabled ISAC paper contrasts it with Gaussian randomization and stresses deterministic tightening and monotonic progress. The near-field ISAC and ISCPT papers argue that higher-rank SDR solutions dilute energy across spatial modes and weaken beam focusing under tight worst-case constraints (Lin et al., 8 Apr 2025, Chen et al., 17 Jul 2025, Liu et al., 2024, Rostamikafaki et al., 15 May 2026).

Theoretical guarantees are more limited than the exactness claims for the preceding convex reformulations. In the continuous-Doppler waveform paper, the SIP-to-SDP conversion over the Doppler interval is exact, but the paper states that formal global optimality or guaranteed rank-one attainment is not universally proven and depends on the problem structure and the tightening schedule. The robust near-field ISAC letter states, based on Theorem 1 in its cited reference, that SROCR converges to a local Karush–Kuhn–Tucker stationary point of the lifted problem and hence of the original design. The ARIS-RSMA paper states convergence to a stationary point of the relaxed problem. Practical convergence is typically supported by feasibility safeguards that halve the tightening increment when infeasibility occurs and by small observed iteration counts in the reported experiments (Lin et al., 8 Apr 2025, Chen et al., 17 Jul 2025, Jin et al., 6 Feb 2026).

Computationally, SROCR inherits the cost of solving a sequence of SDPs. The continuous-Doppler waveform paper uses CVX with MOSEK and reports tractable performance for $F_k=w_k w_k^H$6 up to $F_k=w_k w_k^H$7 and $F_k=w_k w_k^H$8 up to $F_k=w_k w_k^H$9, with a crude per-iteration bound of $$X=\boldsymbol{x}\boldsymbol{x}^H \in \mathbb{H}_+^N$$00 in terms of LMI block dimensions. The robust near-field ISAC letter gives an interior-point complexity characterization in terms of the number and sizes of LMIs and notes that the overall cost scales linearly with the number of SROCR iterations $$X=\boldsymbol{x}\boldsymbol{x}^H \in \mathbb{H}_+^N$$01. The ARIS-RSMA paper uses CVX with SDPT3 or SeDuMi for its SDP blocks. This body of work therefore presents SROCR as a practically deployable but solver-dependent technique: it is strongest when the surrounding convexified model is already well structured, the dominant eigenvector is informative, and the iteration budget is moderate (Lin et al., 8 Apr 2025, Chen et al., 17 Jul 2025, Jin et al., 6 Feb 2026).