Max-Min Secrecy-Rate Optimization

• Max-min secrecy-rate is a fairness-aware approach that maximizes the lowest user secrecy, ensuring secure communications even in the presence of eavesdroppers.
• The problem employs advanced optimization techniques such as DC programming, geometric programming, and barrier methods to handle nonconvex constraints and hardware limitations.
• Practical applications include multi-antenna systems, SWIPT, and IRS/RIS-aided networks, where design choices directly impact system performance and user fairness.

The max-min secrecy-rate problem refers to the design and optimization of physical-layer communication systems to maximize the minimum achievable secrecy rate among multiple users (or user-pairs) in the presence of eavesdroppers. This criterion enforces fairness by ensuring the least-favored user achieves the highest possible secrecy rate, subject to power, hardware, and architectural constraints. The problem is fundamental in secure communications, with realizations in multi-user multi-antenna systems, wiretap channels, SWIPT setups, visible light communications, IRS/RIS-aided systems, and cooperative relay networks.

1. Mathematical Formulation of the Max-Min Secrecy-Rate Problem

The canonical max-min secrecy-rate problem is formulated as: $$\max_{x \in \mathcal{X}} \;\; \min_{k=1,\ldots,K} \; R_{s,k}(x)$$ where $R_{s,k}$ denotes the secrecy rate for user $k$ and $x$ collects all system design variables (power allocations, precoding matrices, beamforming vectors, phase shifts, power-splitting coefficients, etc.).

A prototypical example is the SWIPT (Simultaneous Wireless Information and Power Transfer) wiretap interference channel with two users in presence of a multi-antenna eavesdropper (Kariminezhad et al., 2017):

• Each user $k$ achieves an information rate:

$$R_k = \log\left( 1 + \frac{\eta_k p_k |h_{kk}|^2}{\sigma_k^2 + \eta_k (\varrho_k^2 + \sum_{j\neq k} p_j |h_{kj}|^2)} \right)$$

• The harvested energy constraint at user $k$ is:

$$E_k = (1-\eta_k)\left(\sum_{j=1}^2 p_j |h_{kj}|^2 + \varrho_k^2\right) \geq \psi_k$$

• The eavesdropper’s leakage rate is:

$$R_{E_k} = \log\left(1 + p_k \mathbf{h}_{E,k}^H \Big(\bar{\sigma}_E^2 \mathbf{I} + p_{k+1}\mathbf{h}_{E,(k+1)}\mathbf{h}_{E,(k+1)}^H\Big)^{-1}\mathbf{h}_{E,k}\right)$$

• The secrecy rate for user $k$ is $R_k^s = [R_k - R_{E_k}]^+$.
• The weighted max-min formulation with weights $\alpha_1, \alpha_2, \alpha_1+\alpha_2=1$ is:

$$\max_{p_1,p_2,\eta_1,\eta_2} \min\left\{ \frac{R_1^s}{\alpha_1}, \frac{R_2^s}{\alpha_2} \right\}$$

This framework generalizes naturally to larger networks and different physical architectures.

2. Structural Properties and Nonconvexity

The max-min secrecy-rate problem is generically nonconvex, even under Gaussian inputs and linear channels, because secrecy-rate expressions involve differences of concave (or log-determinant) functions. For example, in Gaussian MIMO wiretap channels,

$$R_s(Q) = \log\det(I + H_b Q H_b^H) - \log\det(I + H_e Q H_e^H)$$

is a difference-of-concave function in the covariance matrix $Q$ (Khabbazibasmenj et al., 2014, Loyka et al., 2015).

In multi-user or SWIPT settings, additional sources of nonconvexity arise from:

• Ratio-of-sum-of-monomials (“signomial”) constraints,
• Discrete variables (e.g., RIS element-user assignments, integer association matrices),
• Nonlinear coupling between multiple hardware parameters (e.g., phase shifts and power allocations),
• Integer and continuous mixed decision variables (Maraqa et al., 8 May 2025).

Consequently, the problem is generally NP-hard. Algorithmic frameworks rely on either structural relaxations, convex approximations, or global optimization via advanced mathematical programming.

3. Algorithmic Approaches

The dominant optimization techniques for the max-min secrecy-rate problem include:

• Difference-of-Convex (DC) Programming: Reformulates secrecy rate maximization as an alternating maximization of concave and convex surrogates, iteratively refining the solution (e.g., alternating eigenspace and eigenvalue updates for transmit covariance matrices) (Khabbazibasmenj et al., 2014).
• Geometric Programming with Single Condensation: For signomial programs (e.g., joint power and PS-coefficient optimization under SWIPT and secrecy constraints), denominators in ratio-of-positives constraints are replaced with monomial lower bounds (via the arithmetic-geometric mean inequality), yielding a sequence of geometric programs solvable via interior-point methods (Kariminezhad et al., 2017).
• Barrier Newton Method for Minimax Reformulation: Introduces an auxiliary minimization (over a noise covariance matrix) to convexify the dual problem, with convergence guaranteed to a global saddle point via primal-dual Newton steps and logarithmic barriers (Loyka et al., 2015).
• Semidefinite Relaxation (SDR) and Charnes–Cooper Transform: Converts quadratic fractional objectives (as in relay or IRS phase shift optimization) into SDPs by lifting the vector variable to a positive semidefinite matrix and normalizing the denominator, enabling tractable solutions up to a relaxation gap (Khordad et al., 2016, Jayarathne et al., 24 Jan 2025).
• Genetic Algorithms for Mixed-Integer Nonlinear Programs (MINLP): For problems with both integer (e.g., RIS assignments) and continuous optimization (e.g., steering angles, power), GAs use population-based search with penalization for constraint violation, providing high-quality solutions at large scale when traditional methods are inapplicable (Maraqa et al., 8 May 2025).
• Fractional Programming: Handles ratio structure in secrecy rates via transformations that enable alternation between transmit power and IRS/beamformer phase optimization with provable monotonic convergence (Jayarathne et al., 24 Jan 2025).

The following table summarizes representative algorithmic classes and application scenarios:

Problem Class	Solution Methodology	Reference
MIMO wiretap (Gaussian input)	DC programming, Newton–barrier	(Khabbazibasmenj et al., 2014, Loyka et al., 2015)
Multi-user/SWIPT wiretap channels	Geometric programming (condensation)	(Kariminezhad et al., 2017)
IRS/RIS-aided multi-user fairness	Alternating optimization (FP+SDR)	(Jayarathne et al., 24 Jan 2025)
Relay networks (2-user, null space BF)	SDR + region splitting	(Khordad et al., 2016)
VLC/RIS with joint integer–continuous	Genetic algorithms	(Maraqa et al., 8 May 2025)

4. Secrecy-Rate Region, Pareto Fronts, and Fairness

The max-min secrecy-rate optimization delineates the Pareto boundary of the achievable secrecy rate region under fairness constraints. The Pareto frontier is typically traced by varying user weights ($\alpha_i$) in the weighted min-ratio objective (Kariminezhad et al., 2017). In the absence of secrecy constraints, rate regions are often convex; however, secrecy constraints can introduce nonconvexity. For instance, in strong interference regimes, secure regions can be strictly nonconvex and time-sharing is required to outperform fixed resource allocations.

Energy harvesting and SWIPT constraints further shrink the achievable region but do not always destroy convexity, as studied in (Kariminezhad et al., 2017). In practical systems, adding fairness (max-min) ensures that no legitimate user is starved, e.g., IRS phase shifts or power allocations are shifted toward disadvantaged links (Jayarathne et al., 24 Jan 2025).

5. Extensions: Eavesdropper Models, Signaling, and Hardware Constraints

• Eavesdropper Processing: In SWIPT interference channels, lower-bounds for secure communication are derived under unconstrained and worst-case linear eavesdropper processing, with the surprising result that for symmetric channels, worst-case eavesdropper optimization does not further degrade secrecy (Kariminezhad et al., 2017).
• Finite-Alphabet Inputs: With practical modulations, the secrecy rate per subchannel saturates at high transmit power, causing traditional covariance allocations to fail (secrecy rate drops to zero). The problem is resolved by imposing upper-power constraints per subchannel (“regularization”), ensuring sustained nonzero secrecy (Vishwakarma et al., 2014).
• Per-Antenna and Energy Constraints: Constraints such as per-antenna power budgets and minimum harvested energy introduce further complexity. These constraints are handled within primal-dual and barrier optimization frameworks (Loyka et al., 2015, Kariminezhad et al., 2017).
• Mixed-Integer Variables: Assignment of RIS elements to users, per-element steering, and random user orientations are modeled as discrete or mixed-integer programs in VLC and RIS-aided systems (Maraqa et al., 8 May 2025).

6. System-Level Insights and Numerical Benchmarks

Empirical studies reveal gains and trade-offs unique to the max-min secrecy regime:

• Multi-user IRS-aided networks demonstrate up to 3.6x improvement in minimal secrecy rate when the IRS is optimally placed, with the greatest effect when the IRS is near vulnerable users—even if the eavesdropper is also nearby (Jayarathne et al., 24 Jan 2025).
• In RIS-aided VLC, the RSMA access scheme yields up to 40–50% higher minimum secrecy rate than NOMA, with dramatic sensitivity of secrecy to photodetector field-of-view and RIS element count (Maraqa et al., 8 May 2025).
• Strong interference degrades the secure rate region’s convexity, motivating time-sharing (Kariminezhad et al., 2017).
• Null-space beamforming at the relay in two-way networks provides the same minimal secrecy as standard schemes with one more relay antenna, or equivalently saves ≈3 dB of SNR (Khordad et al., 2016).

7. Synthesis and Design Guidelines

Max-min secrecy-rate optimization enforces a fairness-aware security regime in multiuser networks, demanding robust joint design of powers, beamformers, RIS/IRS configurations, and signaling. Successive convex relaxations, iterative algorithms (DC, fractional programming, alternating minimization), and evolutionary heuristics dominate the computational landscape depending on the channel model and practical constraints. For IRS/RIS-aided networks, placement and configuration directly target inequities in user secrecy, while SWIPT and finite-alphabet constraints demand new regularization strategies. The max-min secrecy-rate formulation continues to motivate advanced signal processing and mathematical programming approaches in physical-layer secure communications (Kariminezhad et al., 2017, Khabbazibasmenj et al., 2014, Loyka et al., 2015, Maraqa et al., 8 May 2025, Jayarathne et al., 24 Jan 2025, Vishwakarma et al., 2014, Khordad et al., 2016).