Technical Summary: Max-Min Secrecy Rate Optimization for Secure ISAC Networks

1. Problem Statement

The paper addresses the security challenges in Integrated Sensing and Communication (ISAC) systems where multiple communication users (CUs) coexist with multiple untrusted sensing users (SUs). These SUs are modeled as potential eavesdroppers capable of intercepting confidential information intended for the CUs.

The core objective is to design a transmit beamforming strategy at the base station (BS) that maximizes the minimum secrecy rate among all CUs. This "max-min" formulation ensures fairness in secure communication performance. The optimization is subject to two primary physical constraints:

1. Transmit Power Budget: The total power consumed by the BS must not exceed a predefined limit.
2. Sensing Quality: The system must maintain a specific sensing performance, quantified by a beampattern matching error constraint. This ensures the transmitted signal effectively illuminates target directions while suppressing sidelobes.

The resulting optimization problem is highly non-convex due to:

• The secrecy rate expression, which involves the difference between the legitimate user's rate and the maximum eavesdropping rate (a non-concave function).
• The coupling of signal and interference terms in the signal-to-interference-plus-noise ratio (SINR) expressions.
• The non-convex nature of the beampattern matching error constraints.

2. Methodology

To tackle the non-convexity and achieve high-quality solutions, the authors propose two distinct algorithmic approaches: a globally optimal method and a low-complexity alternative.

2.1 Global Optimization via Branch-and-Bound (BB)

The authors first reformulate the problem using Semidefinite Relaxation (SDR) by introducing covariance matrices ($V_k$) and relaxing the rank-one constraints. To handle the remaining non-convex logarithmic constraints arising from the rate expressions, they employ a Branch-and-Bound (BB) framework combined with convex relaxations:

• Convex Envelope: The non-convex constraints involving exponential functions (derived from the logarithmic rate constraints) are relaxed using convex envelopes over specific intervals. This transforms the subproblems into convex optimization problems solvable by standard solvers (e.g., CVXPY).
• Bounding and Branching: The algorithm recursively partitions the feasible region of auxiliary variables. In each iteration, it computes an upper bound (via the convex relaxation) and a lower bound (via a constructed feasible solution).
• Convergence: The process terminates when the gap between the upper and lower bounds falls within a prescribed tolerance ($\epsilon$). The paper provides theoretical proofs (Theorems 2 and 3) guaranteeing that the algorithm converges to an $\epsilon$-optimal global solution within a finite number of iterations.

2.2 Low-Complexity Algorithm via Successive Convex Approximation (SCA)

Recognizing that the BB method's computational complexity grows exponentially with the number of users and antennas, the authors propose a Successive Convex Approximation (SCA) algorithm for practical deployment:

• Linearization: The non-convex constraints are approximated using first-order Taylor expansions (linear lower bounds for the convex exponential functions) around the solution of the previous iteration.
• Iterative Solving: This converts the original problem into a sequence of convex subproblems.
• Convergence: The algorithm iteratively updates the linearization points until the objective value converges. The paper proves that the objective function is non-decreasing and bounded, ensuring convergence to a local optimum.

3. Key Contributions

The paper outlines four primary contributions:

1. System Modeling: Development of a multi-user ISAC framework where untrusted SUs perform simultaneous sensing and eavesdropping, formulated as a max-min secrecy rate problem under beampattern matching constraints.
2. Global Methodology: Proposal of a BB-based algorithm with convex relaxation that guarantees convergence to the global optimum, providing tight upper and lower bounds.
3. Low-Complexity Methodology: Development of an SCA-based algorithm that efficiently finds high-quality local solutions with significantly reduced computational complexity compared to the global method.
4. Performance Analysis: Numerical simulations demonstrating the performance gap between global and local solutions, particularly in high-dimensional settings, and validating the near-optimal performance of the SCA approach.

4. Numerical Results

Simulations were conducted with a BS equipped with 16 antennas serving 3 CUs and sensing 3 untrusted targets under Rician fading channels.

• Convergence: The BB algorithm's upper and lower bounds converged to nearly zero gap after approximately 110 iterations. The SCA algorithm converged much faster (within ~10 iterations) to a solution very close to the global optimum.
• Scalability: As the number of CUs increased (from $K=2$ to $K=4$), the BB algorithm's convergence slowed significantly, requiring over $10^3$ iterations for $K=4$ to reach the same error tolerance. In contrast, the SCA algorithm maintained rapid convergence regardless of the user count.
• Trade-offs:
  • Sensing vs. Security: Tightening the beampattern matching error (requiring higher sensing accuracy) drastically reduced the minimum secrecy rate, as the BS had to allocate most resources to sensing. Conversely, loosening sensing constraints allowed secrecy rates to increase until hitting a saturation point determined by power limits and interference.
  • Antenna Count: Increasing the number of BS antennas (from 10 to 32) significantly improved the minimum secrecy rate. With a large number of antennas (e.g., 32), the system could achieve high sensing accuracy with negligible trade-off in secrecy performance, effectively overcoming the hardware bottleneck.

5. Significance and Claims

The paper claims that its proposed BB algorithm serves as an absolute performance benchmark, establishing the theoretical limit for secrecy rate in secure ISAC networks under the given constraints.

The SCA-based algorithm is presented as a practical solution for real-world deployments. The authors assert that while global optimization is mathematically rigorous, its exponential complexity makes it prohibitive for large-scale systems. The SCA method offers a compelling alternative, achieving near-optimal secrecy performance with significantly lower computational complexity, making it attractive for implementation in large-scale ISAC networks where real-time processing is required.

The work highlights the critical trade-off between sensing quality and communication security, providing a database of results to help system designers determine equilibrium operating points (e.g., specific beampattern error tolerances) that balance radar performance with secure transmission capacity.