Weighted Sum Secrecy Rate (WSSR) Overview

• WSSR is defined as the maximization of a weighted sum of secrecy rates, where each term represents the difference between legitimate and eavesdropper rates.
• It serves as a framework for various secure resource-allocation problems, appearing in models from RSMA confidential broadcast to vehicular networks.
• Algorithmic methods such as successive convexification and alternating optimization are central to solving WSSR and related secrecy-aware problems efficiently.

Searching arXiv for papers related to weighted sum secrecy rate and closely related secrecy-sum optimization formulations. Search query: "weighted sum secrecy rate OR secrecy sum rate weighted arXiv" Weighted Sum Secrecy Rate (WSSR) denotes a secrecy-aware utility in which multiple secrecy-rate terms are combined through nonnegative coefficients, in the canonical multiuser form

$\max \sum_k u_k R_{s,k},$

or, in resource-indexed settings,

$$\max \sum_m\sum_k w_k^m R_k^m,\qquad R_k^m=\bigl[\mathcal C_k^m-\mathcal C_{k,e}^m\bigr]^+.$$

Across recent arXiv literature, this objective appears alongside several closely related formulations: unweighted secrecy sum-rate maximization, weighted sum-rate maximization subject to secrecy-rate constraints, and weighted max-min secrecy-region scalarizations. As a result, WSSR is best understood as one member of a broader class of secrecy-aware resource-allocation problems rather than as a single universally adopted optimization template (Xia et al., 2022, Farooq et al., 2024, Kariminezhad et al., 2017).

1. Canonical meaning and neighboring objective classes

In its most direct form, WSSR maximizes a weighted sum of secrecy rates, where each secrecy term is built from a legitimate rate and an eavesdropper rate. In the standard wiretap-style form used in several adjacent works, the secrecy rate is

$R_{s,i}=\bigl[R_{b,i}-R_{e,i}\bigr]^+.$

In RSMA-based confidential broadcast models, the corresponding user-level secrecy term becomes

$$R_{\mathrm s,k}=\Bigl[R_{\mathrm p,k}-\max_{j\in\mathcal K\backslash\{k\}}R_{k,j}\Bigr]^+,$$

because secrecy is carried only in the private stream, while the common stream is decoded by all users and is therefore non-secret (Xia et al., 2022, Xia et al., 2022).

A recurrent theme in the literature is that many highly relevant papers do not optimize WSSR directly. Instead, they optimize adjacent objectives whose structure is close enough that their models and algorithms transfer almost mechanically to WSSR. In vehicular secrecy-rate maximization, for example, the natural weighted extension is obtained by replacing

$\sum_m\sum_k R_k^m(\mathbf p)$

with

$\sum_m\sum_k w_k^m R_k^m(\mathbf p),$

and the same observation is made for several covariance-design and alternating-optimization frameworks (Farooq et al., 2024).

Objective class	Representative form	Representative sources
Direct WSSR	$\max \sum_k u_k R_{s,k}$ or $\max \sum_{m,k} w_k^m R_k^m$	(Xia et al., 2022, Farooq et al., 2024)
Secrecy-constrained WSR/WESR	$\max \sum_k u_k(C_k+R_{\mathrm p,k})$ s.t. $R_{\mathrm s,k}\ge R_{\mathrm s,k}^{\rm th}$	(Xia et al., 2022, Xia et al., 2022)
Aggregate secrecy utility	$$\max \sum_m\sum_k w_k^m R_k^m,\qquad R_k^m=\bigl[\mathcal C_k^m-\mathcal C_{k,e}^m\bigr]^+.$$0	(Boljević et al., 2024)
Weighted secrecy-region scalarization	$$\max \sum_m\sum_k w_k^m R_k^m,\qquad R_k^m=\bigl[\mathcal C_k^m-\mathcal C_{k,e}^m\bigr]^+.$$1	(Kariminezhad et al., 2017)

This distinction is substantive. In secrecy-constrained weighted sum-rate formulations, secrecy enters as a feasibility requirement rather than as the utility itself. In aggregate secrecy utilities, the optimized quantity may be a sum of legitimate-rate terms minus aggregated leakage terms, without explicit per-user $$\max \sum_m\sum_k w_k^m R_k^m,\qquad R_k^m=\bigl[\mathcal C_k^m-\mathcal C_{k,e}^m\bigr]^+.$$2 secrecy accounting. In weighted secrecy-region methods, the weights act as a rate profile rather than as linear multipliers of secrecy rates.

2. Secrecy models and architectural settings

The systems associated with WSSR and its neighboring objectives are diverse, but they cluster around a few secrecy models. One important class uses internal eavesdroppers. In RSMA-based downlink MISO broadcast channels with confidential messages, each user decodes the common stream, decodes its intended private stream, and then attempts to eavesdrop other users’ private streams. The transmitted signal is

$$\max \sum_m\sum_k w_k^m R_k^m,\qquad R_k^m=\bigl[\mathcal C_k^m-\mathcal C_{k,e}^m\bigr]^+.$$3

the intended private-stream SINR is

$$\max \sum_m\sum_k w_k^m R_k^m,\qquad R_k^m=\bigl[\mathcal C_k^m-\mathcal C_{k,e}^m\bigr]^+.$$4

and the internal-eavesdropping SINR for user $$\max \sum_m\sum_k w_k^m R_k^m,\qquad R_k^m=\bigl[\mathcal C_k^m-\mathcal C_{k,e}^m\bigr]^+.$$5 decoding stream $$\max \sum_m\sum_k w_k^m R_k^m,\qquad R_k^m=\bigl[\mathcal C_k^m-\mathcal C_{k,e}^m\bigr]^+.$$6 is

$$\max \sum_m\sum_k w_k^m R_k^m,\qquad R_k^m=\bigl[\mathcal C_k^m-\mathcal C_{k,e}^m\bigr]^+.$$7

This model makes secrecy fundamentally a private-stream phenomenon, with the strongest internal eavesdropper acting as the secrecy bottleneck (Xia et al., 2022, Xia et al., 2022).

A second class uses external eavesdroppers. In cellular-underlay vehicular networks, a stationary eavesdropper with $$\max \sum_m\sum_k w_k^m R_k^m,\qquad R_k^m=\bigl[\mathcal C_k^m-\mathcal C_{k,e}^m\bigr]^+.$$8 antennas overhears VUE links on reused OFDM resource blocks, with secrecy on RB $$\max \sum_m\sum_k w_k^m R_k^m,\qquad R_k^m=\bigl[\mathcal C_k^m-\mathcal C_{k,e}^m\bigr]^+.$$9 defined as

$R_{s,i}=\bigl[R_{b,i}-R_{e,i}\bigr]^+.$0

In robust MU-MISO downlink beamforming, each legitimate user is paired with a specific single-antenna eavesdropper, and the network secrecy utility is built from

$R_{s,i}=\bigl[R_{b,i}-R_{e,i}\bigr]^+.$1

In multi-user MIMO RCI precoding, the worst-case assumption is even stronger: all unintended users cooperate and form a $R_{s,i}=\bigl[R_{b,i}-R_{e,i}\bigr]^+.$2-antenna eavesdropper for each intended user (Farooq et al., 2024, Zhao et al., 2020, Geraci et al., 2012).

A third class couples secrecy with another physical-layer function. In full-duplex SWIPT, an idle user is simultaneously an energy harvester and a potential eavesdropper, and artificial noise serves two roles at once: degrading interception and supplying RF energy. In SWIPT wiretap interference channels, secrecy rates depend jointly on transmit powers and receiver power-splitting coefficients under EH constraints. In FD-ISAC, malicious targets act as eavesdroppers while sensing constraints are imposed through ISMR. In OIRS-aided VLC, confidential downlink messages coexist with LoS blockage, binary OIRS assignment, IM/DD amplitude constraints, and internal eavesdropping among legitimate users (Wang et al., 2015, Kariminezhad et al., 2017, Boljević et al., 2024, Nguyen et al., 2 Jun 2026).

3. Optimization formulations

Direct WSSR is the most compact formulation: $R_{s,i}=\bigl[R_{b,i}-R_{e,i}\bigr]^+.$3 However, the optimization variables differ sharply across architectures. In vehicular networks they are transmit powers $R_{s,i}=\bigl[R_{b,i}-R_{e,i}\bigr]^+.$4, possibly coupled to RB reuse indicators $R_{s,i}=\bigl[R_{b,i}-R_{e,i}\bigr]^+.$5, although the final solved formulation becomes power-only after simplification. In RSMA they are common-rate allocations, private/common beamformers, auxiliary rate variables, denominator variables, and SINR slack variables. In MIMO secrecy beamforming they are transmit covariance matrices, artificial-noise covariances, and sometimes receive combining vectors. In OIRS-assisted VLC they are the precoder $R_{s,i}=\bigl[R_{b,i}-R_{e,i}\bigr]^+.$6 and a binary assignment tensor $R_{s,i}=\bigl[R_{b,i}-R_{e,i}\bigr]^+.$7. In SWIPT interference channels they are transmit powers and PS coefficients $R_{s,i}=\bigl[R_{b,i}-R_{e,i}\bigr]^+.$8 (Farooq et al., 2024, Xia et al., 2022, Wang et al., 2015, Nguyen et al., 2 Jun 2026, Kariminezhad et al., 2017).

Many influential formulations are WSSR-adjacent rather than direct. In RSMA secure communication with perfect CSIT, the main problem is

$R_{s,i}=\bigl[R_{b,i}-R_{e,i}\bigr]^+.$9

subject to

$$R_{\mathrm s,k}=\Bigl[R_{\mathrm p,k}-\max_{j\in\mathcal K\backslash\{k\}}R_{k,j}\Bigr]^+,$$0

Under imperfect CSIT, the same structure is extended to weighted ergodic sum-rate or weighted average sum-rate with ergodic secrecy constraints. These formulations are explicitly described as secrecy-constrained WSR/WESR rather than direct WSSR (Xia et al., 2022, Xia et al., 2022).

Other neighboring formulations use different scalarizations. In SWIPT wiretap interference channels, secrecy-region boundary points are generated through

$$R_{\mathrm s,k}=\Bigl[R_{\mathrm p,k}-\max_{j\in\mathcal K\backslash\{k\}}R_{k,j}\Bigr]^+,$$1

subject to EH, power, and PS constraints. In FD-ISAC, the optimized quantity is

$$R_{\mathrm s,k}=\Bigl[R_{\mathrm p,k}-\max_{j\in\mathcal K\backslash\{k\}}R_{k,j}\Bigr]^+,$$2

where each term is itself a sum of legitimate-rate contributions minus summed leakage terms to malicious targets. In multi-carrier MIMO with FD jamming, the per-subcarrier secrecy term is

$$R_{\mathrm s,k}=\Bigl[R_{\mathrm p,k}-\max_{j\in\mathcal K\backslash\{k\}}R_{k,j}\Bigr]^+,$$3

and the objective is the sum over subcarriers (Kariminezhad et al., 2017, Boljević et al., 2024, Yang et al., 2018).

A common methodological observation across these models is that inserting positive weights usually changes only the objective coefficients. Several papers state that the same convexification, projection, or alternating-optimization machinery remains valid once $$R_{\mathrm s,k}=\Bigl[R_{\mathrm p,k}-\max_{j\in\mathcal K\backslash\{k\}}R_{k,j}\Bigr]^+,$$4 is replaced by $$R_{\mathrm s,k}=\Bigl[R_{\mathrm p,k}-\max_{j\in\mathcal K\backslash\{k\}}R_{k,j}\Bigr]^+,$$5, because the non-convexity comes from the secrecy-rate structure and interference coupling rather than from the absence of weights (Farooq et al., 2024, Wang et al., 2015, Boljević et al., 2024, Nguyen et al., 2 Jun 2026, Zhao et al., 2020).

4. Algorithmic methods

The dominant algorithmic pattern in WSSR-related work is successive convexification. In secure RSMA with perfect CSIT, non-convex secrecy constraints are handled by SCA through first-order approximations of exponential terms, quadratic-over-linear SINR expressions, and leakage constraints. The resulting convex subproblem is solved by CVX, and the resulting sequence is monotonic nondecreasing and converges to a KKT point or stationary point (Xia et al., 2022, Xia et al., 2022).

Under imperfect CSIT, secure RSMA uses a more elaborate stack: sample average approximation, the rate-WMMSE identity

$$R_{\mathrm s,k}=\Bigl[R_{\mathrm p,k}-\max_{j\in\mathcal K\backslash\{k\}}R_{k,j}\Bigr]^+,$$6

and an alternating procedure in which equalizers and weights are updated in closed form while the precoder is optimized through an inner SCA loop. This produces a joint WMMSE + SCA alternating-optimization algorithm for secrecy-constrained WESR/WASR design (Xia et al., 2022).

Vehicular secrecy-rate maximization combines a standard SCA solver with a low-complexity projected first-order method denoted FISTA or FISTA-L. The gradient structure remains unchanged under weighting except for multiplication by the weights, projection is onto box constraints, and the abstract reports that the FISTA-based approach is at least 300 times faster than the SCA method, albeit with a convergence-runtime trade-off (Farooq et al., 2024).

Several other architectures use different but closely related convexification schemes. Full-duplex SWIPT uses log-exponential reformulation and sequential parametric convex approximation, after first eliminating the uplink receive combiner in closed form. Robust MU-MISO secrecy beamforming uses semidefinite relaxation plus first-order Taylor approximation around deterministic norm-bounded CSI errors. Robust MISO full-duplex wiretap design employs the S-procedure, epigraph reformulation, and bisection over SDP feasibility problems. SWIPT wiretap interference channels use iterative geometric programming via single condensation. Multi-carrier MIMO FD jamming lifts the negative log-det terms by auxiliary matrix variables and then alternates between a convex MAX-DET subproblem and closed-form updates

$$R_{\mathrm s,k}=\Bigl[R_{\mathrm p,k}-\max_{j\in\mathcal K\backslash\{k\}}R_{k,j}\Bigr]^+,$$7

These techniques are all directly reusable in weighted variants because positive weights preserve the same convex or separately convex structure (Wang et al., 2015, Zhao et al., 2020, Vishwakarma et al., 2015, Kariminezhad et al., 2017, Yang et al., 2018).

Two recent application-specific solvers further broaden the algorithmic picture. OIRS-aided VLC secrecy maximization uses alternating optimization with CCCP and first-order Taylor approximations, alternating between OIRS assignment and precoder design. FD-ISAC secrecy maximization uses Iterative Joint Taylor-Block cyclic coordinate descent, with closed-form uplink beamformer updates

$$R_{\mathrm s,k}=\Bigl[R_{\mathrm p,k}-\max_{j\in\mathcal K\backslash\{k\}}R_{k,j}\Bigr]^+,$$8

and a convexified transmit-side subproblem. In both cases, the papers explicitly note that inserting weights into the secrecy objective leaves the main AO/CCCP or Taylor machinery structurally intact (Nguyen et al., 2 Jun 2026, Boljević et al., 2024).

5. Representative domains and empirical behavior

In secure RSMA broadcast channels, the dominant empirical conclusion is that RSMA improves secrecy-aware weighted-sum performance relative to secure MULP. The reported gains are strongest when channels are aligned, when secrecy thresholds increase, and in overloaded regimes with $$R_{\mathrm s,k}=\Bigl[R_{\mathrm p,k}-\max_{j\in\mathcal K\backslash\{k\}}R_{k,j}\Bigr]^+,$$9. As secrecy thresholds grow, power is shifted from the common stream toward private streams, and under imperfect CSIT RSMA achieves better WESR robustness than secure MULP. The same papers also state that NOMA cannot ensure all secrecy constraints in the internal-eavesdropper setting, because one user’s full message may be mapped to a stream decoded by others (Xia et al., 2022, Xia et al., 2022).

In vehicular secrecy-rate maximization, the empirical trends are resource-stress trends: secrecy rate decreases as the number of eavesdropper antennas $\sum_m\sum_k R_k^m(\mathbf p)$0 increases and as the number of VUE pairs increases, especially at low speed where inter-VUE interference is stronger. The numerical comparison between SCA and FISTA/FISTA-L shows that the first-order methods nearly converge to the same solution as SCA, sometimes about $\sum_m\sum_k R_k^m(\mathbf p)$1 worse, while being hundreds to thousands of times faster in the reported scenarios (Farooq et al., 2024).

In full-duplex SWIPT and FD-ISAC, secrecy performance is governed by the tension between artificial noise and self-interference. In FD-SWIPT, the proposed FD scheme outperforms HD by about $\sum_m\sum_k R_k^m(\mathbf p)$2 in one reported scenario, while secrecy decreases as the harvested-energy requirement rises and as residual SI grows. In FD-ISAC, IJTB reaches stability within approximately $\sum_m\sum_k R_k^m(\mathbf p)$3 iterations, and the benchmarks’ secrecy performance approaches zero as the eavesdropper distance reaches $\sum_m\sum_k R_k^m(\mathbf p)$4 meters. The paper also reports about $\sum_m\sum_k R_k^m(\mathbf p)$5 bps/Hz for IJTB versus $\sum_m\sum_k R_k^m(\mathbf p)$6 bps/Hz for isotropic no-AN at $\sum_m\sum_k R_k^m(\mathbf p)$7 m in one setting (Wang et al., 2015, Boljević et al., 2024).

In OIRS-aided VLC, increasing the number of OIRS units monotonically improves SSR. For the one-blocked-user scenario, increasing the number of OIRS units from $\sum_m\sum_k R_k^m(\mathbf p)$8 to $\sum_m\sum_k R_k^m(\mathbf p)$9 raises SSR from roughly $\sum_m\sum_k w_k^m R_k^m(\mathbf p),$0 to $\sum_m\sum_k w_k^m R_k^m(\mathbf p),$1 bps/Hz; for the two-blocked-user scenario, it rises from roughly $\sum_m\sum_k w_k^m R_k^m(\mathbf p),$2 to $\sum_m\sum_k w_k^m R_k^m(\mathbf p),$3 bps/Hz. In multi-carrier MIMO FD jamming, optimized jamming improves secrecy but the gain saturates with jamming power and can reverse under poor SI cancellation if jamming is not carefully optimized (Nguyen et al., 2 Jun 2026, Yang et al., 2018).

Older but still structurally relevant results show the same pattern in other regimes. In fading MAC-WT without instantaneous Eve CSI, cooperative jamming boosts secrecy sum-rate significantly, and at high SNR the secrecy sum-rate with CJ and no Eve CSI exceeds the secrecy sum-rate without CJ and with full Eve CSI. In robust MU-MISO secrecy beamforming, the SDR + SCA design outperforms both the lower-complexity ZF-based design and an SLNR-based algorithm, while the ZF construction becomes attractive when $\sum_m\sum_k w_k^m R_k^m(\mathbf p),$4 and can be combined with water-filling-like power allocation (Shah et al., 2011, Zhao et al., 2020).

6. Secrecy regions, weighted decompositions, and conceptual boundaries

A central conceptual boundary is the distinction between WSSR as an optimization objective and weighted expressions that arise analytically. In optimal source selection with unreliable backhaul, the high-SNR ergodic secrecy rate can be written as

$\sum_m\sum_k w_k^m R_k^m(\mathbf p),$5

where the weights $\sum_m\sum_k w_k^m R_k^m(\mathbf p),$6 are induced by backhaul activation probabilities and source-selection/eavesdropper competition. That weighted sum is an analytical decomposition of one ESR, not a designer-chosen weighted secrecy utility across users or flows (Kundu et al., 2021).

A second boundary concerns weighted sum versus weighted max-min. In SWIPT wiretap interference channels, the secure rate region may become non-convex as interference grows, and time-sharing enlarges the secrecy region in strong-interference cases. In that setting, weighted max-min or rate-profile scalarization is used specifically to trace Pareto-boundary points. This suggests that linear WSSR and weighted max-min secrecy-region methods are not interchangeable, especially when the secrecy region is non-convex (Kariminezhad et al., 2017).

A third boundary concerns aggregate secrecy utility versus strict per-user secrecy accounting. In FD-ISAC, the optimized quantity is

$\sum_m\sum_k w_k^m R_k^m(\mathbf p),$7

with aggregated leakage sums over eavesdroppers rather than a userwise $\sum_m\sum_k w_k^m R_k^m(\mathbf p),$8 secrecy model. In secrecy-constrained RSMA WSR/WESR, the optimized utility is weighted total user rate while secrecy enters through thresholds. Both are highly relevant to WSSR methodology, but neither is identical to direct weighted secrecy-rate maximization (Boljević et al., 2024, Xia et al., 2022, Xia et al., 2022).

This suggests that any precise WSSR statement must specify at least four modeling choices: the secrecy definition itself, the eavesdropper model, the object being weighted, and the scalarization type. The literature surveyed here covers per-user secrecy rates, per-subcarrier secrecy rates, aggregate UL/DL secrecy utilities, and secrecy-region boundary methods; internal eavesdroppers, passive external eavesdroppers, and cooperating unintended users; weights on secrecy terms, weights on aggregate utilities, and rate-profile parameters; and solution methods ranging from SCA and SPCA to WMMSE, CCCP, MAX-DET, GP, SDR, and SDP-based robustification (Farooq et al., 2024, Yang et al., 2018, Vishwakarma et al., 2015).