Max-Min Secrecy Rate Optimization for Secure ISAC Networks: Global Optimization and Low-Complexity Algorithm

Dang Y Hoang; Hoang-Lai Pham; Thanh-Nha To; Trung Quang Pham; Tuan Anh Pham

arxiv: 2606.13582 · v1 · pith:A3ACTHFFnew · submitted 2026-06-11 · 📡 eess.SP · cs.IT· math.IT

Max-Min Secrecy Rate Optimization for Secure ISAC Networks: Global Optimization and Low-Complexity Algorithm

Thanh-Nha To , Trung Quang Pham , Dang Y Hoang , Hoang-Lai Pham , Tuan Anh Pham This is my paper

Pith reviewed 2026-06-27 05:37 UTC · model grok-4.3

classification 📡 eess.SP cs.ITmath.IT

keywords secure ISACmax-min secrecy ratebranch-and-boundsuccessive convex approximationsemidefinite relaxationbeampattern matching erroruntrusted sensing users

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The pith

Branch-and-bound solves the max-min secrecy rate problem to global optimality in secure ISAC networks while a successive convex approximation method reaches near-optimal performance at lower cost.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets up a max-min secrecy rate optimization for an ISAC system where multiple communication users share the channel with untrusted sensing users that could eavesdrop. The problem includes a total power limit and beampattern matching error constraints to keep sensing quality acceptable. Semidefinite relaxation converts the non-convex secrecy-rate and sensing expressions into a form that admits a branch-and-bound search for the global solution within a chosen accuracy. A successive convex approximation routine is then derived that solves a sequence of convex subproblems and converges to a local point. The resulting pair of methods supplies both an exact benchmark and a practical low-complexity alternative for systems that must balance secrecy fairness with sensing requirements.

Core claim

The branch-and-bound algorithm, built on semidefinite relaxation and successive convex relaxations, returns the globally optimal max-min secrecy rate subject to the transmit power budget and the beampattern matching error constraints; the successive convex approximation algorithm produces a locally optimal solution whose secrecy performance is nearly identical but whose run time is substantially smaller.

What carries the argument

Branch-and-bound framework applied after semidefinite relaxation of the non-convex secrecy-rate and beampattern constraints.

If this is right

The branch-and-bound solution serves as an exact performance benchmark against which other secure ISAC algorithms can be measured.
The SCA algorithm converges to a stationary point of the relaxed problem and therefore supplies a usable operating point under tight latency constraints.
Maximizing the minimum secrecy rate directly enforces fairness across communication users even when sensing users act as potential eavesdroppers.
Beampattern matching error constraints translate the sensing-quality requirement into a form that remains compatible with the secrecy-rate objective.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The global optimum obtained by branch-and-bound can be used to quantify the sub-optimality gap of any future heuristic proposed for larger ISAC deployments.
Because the SCA iterates are convex, the same successive-approximation template could be reused for related ISAC problems that add mobility or imperfect channel state information.
If real radar performance deviates from the beampattern matching error metric, the reported secrecy rates would need recalibration against measured range or angle estimation error.

Load-bearing premise

The semidefinite relaxation together with the convex relaxations inside branch-and-bound produces solutions whose gap to the original non-convex problem is negligible.

What would settle it

On a small instance with two communication users and two sensing users, run both the branch-and-bound solver and the SCA solver to completion and check whether the secrecy-rate values differ by more than a few percent while both satisfy the beampattern error limit.

Figures

Figures reproduced from arXiv: 2606.13582 by Dang Y Hoang, Hoang-Lai Pham, Thanh-Nha To, Trung Quang Pham, Tuan Anh Pham.

**Figure 1.** Figure 1: Convex envelope of the non-convex set D[l,u] . 3.3 Proposed BB Algorithm To globally solve problem (15), we adopt a BB framework based on the convex relaxation (17). Let B = [a (l) , a (u) ] × [b (l) , b (u) ] denote a box (hyper-rectangle) in the space of auxiliary variables a, b. The BB algorithm proceeds by recursively partitioning B and computing upper and lower bounds over each subregion. Initializat… view at source ↗

**Figure 2.** Figure 2: Convergence behavior of the proposed BB and SCA algorithms. yield nearly identical objective values, indicating that the SCA method is relatively insensitive to initialization and provides a low-complexity alternative [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Figure 3: Relative error of the proposed BB algorithm and SCA algorithm. -26 -24 -22 -20 -18 -16 -14 -12 -10 Max beampattern error (dB) 0 2 4 6 8 10 12 14 Min secrecy rate (bps/Hz) K = 2 K = 3 K = 4 K = 6 K = 8 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

**Figure 4.** Figure 4: Min secrecy rate versus max beampattern error under different number of CUs [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

**Figure 5.** Figure 5: Min secrecy rate versus the number of BS antennas under different max beampattern error. 6 Conclusions In this paper, we investigate a secure ISAC system with unreliable SUs acting as potential eavesdroppers. We construct a max-min secrecy rate optimization problem under constraints on transmit power and sensing quality. To solve this non-convex problem, we develop a global optimization framework based on … view at source ↗

read the original abstract

In this paper, we investigate a secure integrated sensing and communication (ISAC) system in which multiple communication users (CUs) coexist with multiple untrusted sensing users (SUs) that may eavesdrop on the confidential information intended for the CUs. To promote security fairness among users, we formulate a max-min secrecy rate optimization problem subject to a transmit power budget and sensing quality requirements characterized by beampattern matching error constraints. The resulting design problem is highly non-convex due to the secrecy rate expressions and non-convex sensing constraints. To address these challenges, we first reformulate the problem using semidefinite relaxation (SDR). Based on the reformulated problem, we develop a branch-and-bound (BB) framework combined with convex relaxations to obtain the globally optimal solution within a prescribed accuracy. To further reduce computational complexity, we propose a low-complexity algorithm based on successive convex approximation (SCA), which iteratively solves a sequence of convex subproblems and converges to a local solution. Numerical results demonstrate that the proposed BB algorithm achieves the global optimum and provides a benchmark for performance evaluation. Moreover, the proposed SCA-based algorithm attains near-optimal secrecy performance with significantly lower computational complexity, making it attractive for practical ISAC deployments.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Standard SDR plus branch-and-bound for global max-min secrecy rate in ISAC with untrusted SUs, paired with an SCA fallback that gets close in the numerics.

read the letter

This paper takes the usual SDR reformulation, wraps it in a branch-and-bound procedure to claim global optimality for the max-min secrecy rate, and adds a successive convex approximation routine that runs faster. The setting adds untrusted sensing users that can eavesdrop on the communication users, with both total power and beampattern matching error constraints.

What stands out is the direct comparison: the BB solution serves as a benchmark, and the SCA version is shown to land close while cutting complexity. That numerical evidence is the practical takeaway for anyone who needs to trade secrecy fairness against sensing quality under a power budget.

The soft spots are the standard ones for this class of work. Secrecy-rate expressions after SDR are not automatically convex, and the beampattern error is only a surrogate; if rank-1 solutions do not hold or the surrogate deviates from actual sensing metrics such as CRLB, the reported optimality gap could be larger than claimed. The abstract states that the BB reaches the global optimum and SCA is near-optimal, but without seeing the explicit rank checks or the size of the duality gaps in the results, it is hard to judge how tight the relaxations actually are in the tested regimes.

The paper is aimed at researchers who already work on convex optimization for secure ISAC and want a ready benchmark plus a low-complexity alternative. It is coherent on its own terms and uses established tools without circularity, so it deserves a serious referee who can verify the tightness claims against the full derivations and figures.

Referee Report

2 major / 1 minor

Summary. The paper formulates a max-min secrecy rate optimization problem for a secure ISAC system with multiple CUs and untrusted SUs that may eavesdrop, subject to a transmit power budget and beampattern matching error constraints for sensing quality. The resulting non-convex problem is addressed via SDR reformulation, followed by a branch-and-bound framework with convex relaxations to obtain a globally optimal solution within prescribed accuracy, and a low-complexity SCA algorithm that iteratively solves convex subproblems to a local solution. Numerical results are presented to show that BB achieves the global optimum (serving as benchmark) while SCA attains near-optimal secrecy performance at lower complexity.

Significance. If the SDR tightness and beampattern surrogate are validated to produce negligible gaps, the work supplies a useful global benchmark and practical low-complexity method for fairness-oriented secure ISAC designs; the combination of global and approximate solvers is a standard but valuable contribution in this domain when accompanied by reproducible validation.

major comments (2)

[Abstract] Abstract (and BB algorithm description): the central claim that the BB algorithm achieves the global optimum of the original problem rests on the assertion that SDR plus the convex relaxations inside the branch-and-bound framework produce bounds whose gap to the non-convex max-min secrecy-rate objective (involving differences of log(1+SINR) terms) and the quadratic beampattern matching error constraints is negligible. No explicit rank-1 recovery guarantee, optimality-gap analysis, or comparison against the original non-convex formulation is referenced; without this, the global-optimality statement cannot be verified from the given material.
[Abstract] Abstract (sensing constraints): the beampattern matching error is adopted as the sole sensing-quality metric, yet its relationship to standard sensing figures of merit (CRLB, detection probability, or estimation variance) is not quantified. If the surrogate deviates materially from the actual sensing requirement, both the global optimum reported by BB and the near-optimality claimed for SCA become surrogate-specific rather than directly applicable to practical ISAC sensing performance.

minor comments (1)

The abstract states that the SCA algorithm converges but supplies no convergence rate, monotonicity argument, or stopping criterion; a brief reference to the relevant theorem or lemma would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below and indicate the revisions planned for the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract (and BB algorithm description): the central claim that the BB algorithm achieves the global optimum of the original problem rests on the assertion that SDR plus the convex relaxations inside the branch-and-bound framework produce bounds whose gap to the non-convex max-min secrecy-rate objective (involving differences of log(1+SINR) terms) and the quadratic beampattern matching error constraints is negligible. No explicit rank-1 recovery guarantee, optimality-gap analysis, or comparison against the original non-convex formulation is referenced; without this, the global-optimality statement cannot be verified from the given material.

Authors: The branch-and-bound procedure is applied to the SDR-relaxed formulation to compute its global optimum to within a prescribed accuracy. Numerical experiments in the manuscript show that the resulting covariance matrices are rank-1 in the evaluated scenarios, permitting recovery of feasible beamformers for the original problem. We agree, however, that an explicit tightness analysis, rank-1 recovery guarantee, or quantitative optimality-gap bound relative to the original non-convex problem is absent. In the revision we will add a dedicated subsection that (i) states the global optimum is obtained for the SDR problem, (ii) reports the observed rank-1 property together with the empirical gap to the original objective, and (iii) clarifies the abstract and introduction accordingly. revision: yes
Referee: [Abstract] Abstract (sensing constraints): the beampattern matching error is adopted as the sole sensing-quality metric, yet its relationship to standard sensing figures of merit (CRLB, detection probability, or estimation variance) is not quantified. If the surrogate deviates materially from the actual sensing requirement, both the global optimum reported by BB and the near-optimality claimed for SCA become surrogate-specific rather than directly applicable to practical ISAC sensing performance.

Authors: The beampattern matching error is a widely adopted surrogate in the ISAC literature because it directly enforces similarity between the transmitted covariance and a prescribed sensing beampattern, which is relevant for applications that require controlled angular illumination. A quantitative mapping to CRLB, detection probability, or estimation variance is not derived in the present work. We will revise the manuscript to include a short discussion of this modeling choice, its relation to practical sensing metrics, and an explicit statement that the reported performance is with respect to the adopted surrogate. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper formulates a max-min secrecy rate problem and applies standard SDR reformulation followed by branch-and-bound with convex relaxations for global optimality and SCA for local solutions. No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations, uniqueness theorems imported from prior author work, ansatzes smuggled via citation, or renamings of known results appear in the provided text. The derivation chain consists of applying established convex optimization techniques to the stated non-convex problem, remaining self-contained without reduction of claims to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard domain assumptions in convex optimization and wireless modeling. No free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption Semidefinite relaxation yields a solution whose optimality gap is negligible for the secrecy-rate and beampattern constraints in this problem class.
Invoked when the problem is reformulated using SDR prior to applying branch-and-bound.

pith-pipeline@v0.9.1-grok · 5771 in / 1193 out tokens · 33933 ms · 2026-06-27T05:37:27.938472+00:00 · methodology

discussion (0)

Reference graph

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Bao, J., Cao, Y., Tang, J., Xu, L., Zhao, N., Wang, X.: Secrecy energy eﬃciency enhancement of active irs aided miso networks. IEEE Transactions on Vehicular Technology 73(5), 7345–7350 (2024)

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Ren, Z., Xu, J., Qiu, L., Ng, D.W.K.: Secure cell-free integrated sensing and communication in the presence of information and sensing eavesdroppers. IEEE Journal on Selected Areas in Communications 42(11), 3217–3231 (2024)

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In: 2023 21st International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt)

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2025