Differential Evolution with One-in-One Representation

• The paper presents DEO as a novel optimization strategy using one-in-one representation and adaptive penalties to reduce the search dimensionality for movable antenna positioning.
• DEO is integrated within a two-layer iterative framework that couples an inner alternating-optimization for beamforming and RIS phase design with an outer loop for antenna placement.
• The approach effectively enhances secrecy rates in RIS-aided cell-free systems, demonstrating up to a 12.4% performance gain over conventional benchmarks.

Searching arXiv for the cited paper to ground the article. Differential Evolution with One-in-One Representation (DEO) is a population-based optimization strategy introduced, in the cited setting, as the outer-layer solver for movable-antenna position design in an RIS-aided cell-free symbiotic radio system with secure transmission objectives. In the formulation reported in "Secure Transmission for Cell-Free Symbiotic Radio Communications with Movable Antenna: Continuous and Discrete Positioning Designs" (Lyu et al., 9 Aug 2025), DEO is coupled to an inner alternating-optimization procedure and is characterized by two explicit design features: one-in-one representation and an adaptive penalty mechanism. The method is used to optimize movable antenna (MA) positions under secrecy-rate maximization, secondary-transmission constraints, MA movement-region constraints, and minimum antenna-spacing constraints. The same work further extends the DEO framework to a discrete-position setting through mapping and determination operations, while also presenting an alternative alternating-optimization framework with lower computational complexity (Lyu et al., 9 Aug 2025).

1. Definition and optimization setting

In the reported system model, there are $M$ distributed access points (APs), each AP has $N$ movable antennas, and the network serves $N_p$ primary users (PUs), $N_s$ secondary users (SUs), and $N_e$ eavesdroppers, with one RIS containing $N_\varepsilon$ passive reflecting elements (Lyu et al., 9 Aug 2025). The APs transmit primary symbols $s_k$ to the PUs, while the RIS backscatters or reuses the primary signal to deliver a secondary symbol $c$ to the SUs. Eavesdroppers attempt to intercept the confidential primary information.

For continuous MA positions, AP $m$'s MA locations are collected as

$$\bm{t}_m = [\bm{t}_{m,1}^T,\ldots,\bm{t}_{m,N}^T]^T,\qquad \bm{t}_{m,n}=[x_{m,n},y_{m,n},z_{m,n}]^T.$$

The AP-to-RIS channel and AP-to-node channels depend on these positions: $N$0

$N$1

The RIS phase-shift matrix is

$N$2

and AP $N$3's transmitted signal is

$N$4

The optimization criterion is the minimum secrecy rate over primary users and eavesdroppers. With achievable rates

$N$5

the secrecy rate of primary user $N$6 against eavesdropper $N$7 is

$N$8

and the overall objective is

$N$9

For the continuous-position case, the problem is formulated with auxiliary variable $N_p$0 as

$N_p$1

2. Two-layer iterative structure

The cited work places DEO inside a two-layer iterative framework because the continuous-position problem is highly non-convex and couples beamforming, RIS phases, and antenna positions (Lyu et al., 9 Aug 2025). The decomposition separates the continuous MA-position search from the optimization of beamforming and RIS coefficients.

In the inner layer, for fixed MA positions $N_p$2, the method solves

$N_p$3

This stage is handled by alternating optimization (AO): beamformers are optimized with fixed RIS coefficients, RIS phases are optimized with fixed beamformers, and the process is iterated until convergence.

In the outer layer, after obtaining $N_p$4 and $N_p$5 from the inner layer, the method optimizes MA positions through

$N_p$6

This outer search is performed by DEO.

This layering is central to the reported role of DEO. It is not a standalone end-to-end solver for all variables; rather, it is an outer-loop mechanism for antenna-position search that repeatedly relies on inner-layer optimization to evaluate candidate populations. A plausible implication is that DEO is tailored here to the geometry-dependent part of the design space, while AO handles the continuous signal-processing blocks.

3. One-in-one representation and DEO population dynamics

The paper explicitly identifies one-in-one representation as a defining property of DEO (Lyu et al., 9 Aug 2025). Instead of encoding the full MA configuration as one high-dimensional particle, DEO treats each MA as one particle: $N_p$7 where each particle is the 3D position of a single antenna,

$N_p$8

The population size is therefore effectively $N_p$9, one particle per antenna. The reported motivation is that this reduces dimensionality and improves search efficiency.

The fitness function for the current MA population $N_s$0 is

$N_s$1

where $N_s$2 is the secrecy-rate objective value returned by the inner-layer optimization, $N_s$3 is a large penalty weight, $N_s$4 is the set of MA pairs violating the minimum-distance constraint, and

$N_s$5

This construction makes the fitness simultaneously depend on the achieved secrecy objective and the total severity of spacing violations.

For mutation, DEO generates a donor vector for particle $N_s$6 as

$N_s$7

where $N_s$8 are distinct randomly selected particles and $N_s$9 is the scaling factor. Binomial crossover then produces the trial vector $N_e$0: $N_e$1 where $N_e$2, $N_e$3 is the crossover probability, and $N_e$4 ensures that at least one donor component is retained.

Selection is performed by replacing one random particle with $N_e$5 and accepting the update only if the resulting population fitness improves: $N_e$6 Otherwise the original population is preserved.

4. Constraint handling and interaction with the inner optimization

Constraint handling in DEO is inseparable from the secrecy-rate maximization architecture used in the cited work. Boundary violations in MA coordinates are handled by coordinate-wise clipping: $N_e$7 Spacing constraints are not enforced by direct projection; instead, they are incorporated through the DEO penalty term described above. This makes the outer search feasibility-aware without removing all infeasible candidates a priori.

The inner layer uses SDR and SCA for beamforming optimization. The beamforming variables are lifted as

$N_e$8

The secrecy constraint is decomposed as a difference of concave terms,

$N_e$9

with $N_\varepsilon$0 and $N_\varepsilon$1 linearized by first-order Taylor approximation.

RIS optimization similarly introduces

$N_\varepsilon$2

and enforces the rank-one condition through

$N_\varepsilon$3

which is incorporated into the objective via the penalty term

$N_\varepsilon$4

The stopping rule for the DEO outer loop is a preset maximum iteration number $N_\varepsilon$5, with the nested convergence behavior also influenced by the inner-layer stopping criteria. This suggests that the effective computational profile of DEO in this application depends not only on population evolution but also on the cost of repeatedly solving the inner AO subproblem.

5. Discrete-position extension

The same paper extends DEO from continuous MA positioning to a discrete setting in which each MA must choose one location from a finite candidate set

$N_\varepsilon$6

A binary selection matrix $N_\varepsilon$7 is used, with each column selecting one candidate position for one MA (Lyu et al., 9 Aug 2025). The corresponding optimization problem is

$N_\varepsilon$8

In the extended DEO framework, the initial population is defined over binary-selection encodings: $N_\varepsilon$9 Mutation and crossover are applied in the same DEO manner, but on $s_k$0-dimensional particles.

Because DEO offspring are real-valued, the paper adds a mapping operation: $s_k$1 This sigmoid map converts the offspring entries to probabilities. A subsequent determination operation transforms each column into a one-hot assignment by setting the largest entry in a column to 1 and all others to 0, while excluding already used row indices from later columns so that each discrete position is assigned to at most one MA. The resulting encoding satisfies

$s_k$2

The minimum-distance constraint in the discrete case is enforced through

$s_k$3

If a DEO-generated discrete assignment violates this condition, the candidate is discarded. In this form, DEO no longer searches directly over Euclidean coordinates alone; it searches over a binary-coded assignment structure, then uses mapping and determination to restore feasibility with respect to the combinatorial placement model.

6. Alternative AO framework, complexity, and reported numerical behavior

For the discrete-position case, the paper also introduces an alternative AO iterative framework intended to reduce complexity (Lyu et al., 9 Aug 2025). Instead of applying DEO to the discrete MA-position block, the problem is decomposed into three subproblems: optimization of beamforming $s_k$4, optimization of RIS phases $s_k$5, and optimization of the discrete position matrices $s_k$6.

In the discrete-position subproblem, lifted variables are introduced through

$s_k$7

with the rank-one condition enforced via

$s_k$8

The binary constraint is relaxed; after solving the convex surrogate, the eigenvector together with the same mapping and determination operations is used to recover the binary matrices $s_k$9.

The complexity expressions given in the paper are

$c$0

and

$c$1

with the conclusion

$c$2

The stated reason is that DEO-based continuous or discrete outer search is more expensive because every DEO iteration requires repeated solution of the inner AO subproblem, whereas the AO discrete framework optimizes the three blocks directly in a single loop.

The numerical findings reported for the same system characterize both convergence and performance. The DEO-based two-layer framework converges in both continuous and discrete cases, with secrecy rate increasing over iterations and then stabilizing. The discrete AO framework also converges and reaches a stable value after about 7 iterations. In terms of performance, continuous MA positioning outperforms discrete MA positioning; at $c$3 dBm, the proposed continuous-position scheme with DEO achieves about 12.4% improvement over the FPA benchmark, and continuous-position DEO is about 6.9% better than discrete-position DEO at the same transmit-power level. The discrete AO framework remains close to the discrete DEO method, with performance about 1.55% lower on average at $c$4 dBm.

These results delimit the practical role of DEO in the reported work. DEO is effective in the continuous-position regime and remains applicable to the discrete-position regime after binary-encoding extensions, but the paper ultimately identifies the AO framework as the lower-complexity alternative for the discrete case.

7. Methodological significance and scope

Within the cited study, DEO is significant not as a generic reformulation of differential evolution, but as a specific structural adaptation for movable-antenna optimization under coupled physical-layer security and symbiotic-radio constraints (Lyu et al., 9 Aug 2025). The paper attributes its effectiveness to four factors: reduction of MA search dimension through one-in-one representation, effective exploration of continuous antenna space, a penalty mechanism that steers the search away from infeasible antenna placements, and natural integration with the inner AO secrecy-rate maximization.

A common misunderstanding would be to treat DEO here as the sole optimization engine. The reported formulation does not support that interpretation. Instead, DEO is embedded in a broader solver stack that includes AO, SCA, SDR, and penalty-based rank-one enforcement. Another possible misconception is that the discrete-position extension simply discretizes the continuous algorithm. The description in the paper indicates a more substantial modification: DEO is extended to binary-coded assignments and augmented with sigmoid mapping, determination operations, and explicit candidate rejection when minimum-distance requirements are violated.

More broadly, the reported formulation suggests that one-in-one representation is the defining conceptual contribution of DEO in this application. A plausible implication is that the method is especially suited to optimization problems in which a large joint spatial configuration can be decomposed into many low-dimensional entities while still being evaluated through a global objective. In the cited work, that objective is the minimum secrecy rate under secondary-transmission and geometric constraints, and DEO serves precisely as the mechanism for navigating that decomposed search space.