Discover Fast Power Allocation Solution for Multi-Target Tracking via AlphaEvolve Evolution

Abstract: Efficient radar resource allocation is a fundamental yet computationally challenging problem, as optimal solutions typically require iterative optimization with high complexity. Motivated by the need for real-time scheduling, robust generalization, and low data dependency, this paper proposes a novel paradigm that leverages LLM-guided evolutionary search (AlphaEvolve) to autonomously discover a closed-form power allocation solution for multi-target tracking. The approach encodes high-dimensional radar states into physically inspired features, then evolves a compact and interpretable scoring function, which is transformed to feasible power allocations via a deterministic constraint-satisfying transformation. Extensive experiments demonstrate that the discovered closed-form solution achieves near-optimal tracking accuracy (average relative performance loss of only $1.51\%$), reliable generalization across diverse scenarios and target counts, and over three orders of magnitude speedup compared to conventional iterative solvers. These results highlight the potential of LLM-guided symbolic search to revolutionize not only radar resource management but also broader classes of engineering optimization problems.

• The paper demonstrates a near-optimal allocation method that reduces relative performance loss to 1.51% compared to iterative optimization methods.
• It employs LLM-guided symbolic regression to derive compact, closed-form scoring functions with O(N) computational complexity for real-time tracking.
• Empirical results validate robust generalization up to 200 targets while maintaining interpretability and efficiency in diverse radar scenarios.

LLM-Guided Symbolic Evolution for Radar Power Allocation: An Expert Analysis

Introduction and Problem Formulation

The paper "Discover Fast Power Allocation Solution for Multi-Target Tracking via AlphaEvolve Evolution" (2605.01794) addresses the fundamental challenge of real-time radar resource allocation in multi-target tracking. This problem is characterized by strong combinatorial complexity, physical constraints, and the need for robustness and low-latency decision making. The canonical objective is to allocate the total radar transmit power among $N$ dynamically varying targets to minimize the weighted sum of their Bayesian Cramér-Rao lower bound (BCRLB) tracking errors, subject to both sum and per-target power constraints.

While the underlying optimization $\mathcal{P}$ is convex, closed-form solutions are impractical due to the intractability of matrix inversion dependencies in the objective. Traditional iterative approaches (IPM, sequential quadratic programming, particle swarm, etc.) exhibit $\mathcal{O}(N^3)$ runtime per iteration and are unsuitable for high-throughput, real-time adaptive radar operation.

Approach: Physics-Guided Symbolic Evolution

The paper proposes a paradigm shift from iterative optimization and black-box deep learning approximations to a white-box, closed-form solution discovery leveraging recent advances in LLM-augmented symbolic regression—concretely the AlphaEvolve framework. The process involves:

• Extracting a compact, physically-grounded feature vector for each target, emphasizing the demand factor $D_i$, baseline marginal benefit $M_{\text{base},i}$, and their scenario-normalized variants.
• Guiding AlphaEvolve's LLM-centric evolutionary search—subject to an operator-constrained algebraic expression library—on these features to optimize for generalizable, parsimonious scoring functions mapping scenario features to power priority scores, balancing approximation error, out-of-distribution robustness, and formula complexity.

  Figure 1: AlphaEvolve framework for radar power allocation.

The closed-form scoring functions discovered are then deterministically mapped to feasible power allocations via a clamped, normalized reweighting to impose physical constraints without iteration.

Feature Engineering and Symbolic Search

A notable methodological contribution is the systematic construction of interpretable, tractable feature vectors informed by KKT condition analysis. Features reduce high-dimensional radar state matrices to domain-relevant statistics, such as demand factors, marginal benefits at different allocation operating points, and global statistics (means, variances, maxima over the scenario), while intentionally omitting information not impacting first-order allocation sensitivity.

AlphaEvolve is tasked, via cascaded evaluation, to evolve scoring functions $S_i = f(\mathbf{u}_i)$ that map the compact features to power priorities. Strong multi-objective filtering penalizes lack of robustness, overfitting, and formulaic complexity. The resulting function selected is:

$$S_i = \max\left((\eta_i \zeta_i)^{0.495}, 10^{-6}\right)$$

where $\eta_i$ and $\zeta_i$ are normalized versions of the absolute demand factor and baseline marginal benefit, respectively.

Numerical Results and Empirical Performance

Static Approximation and Robustness

Numerical evaluation on $10^4$ randomly generated scenarios with $\mathcal{P}$0 compared the AlphaEvolve formula, high-SNR approximation, and uniform allocation. The average relative performance loss (versus IPM optimality) was:

• Uniform: $\mathcal{P}$1
• High-SNR approximation: $\mathcal{P}$2
• Discovered formula: $\mathcal{P}$3

with the latter exhibiting minimal variance and absence of pathological outliers.

Figure 2: Absolute accuracy and long-tail robustness.

Generalization to Large Scale

Stress-testing for $\mathcal{P}$4 up to $\mathcal{P}$5 targets shows the error band for the discovered formula firmly bounded ($\mathcal{P}$6), with narrow percentile bands across all scales, in sharp contrast to significant degradation in the baselines.

Figure 3: Generalization under scale expansion.

Computational Efficiency

The closed-form solution achieves $\mathcal{P}$7 vectorized complexity, consistently operating below $\mathcal{P}$8s—even for $\mathcal{P}$9—compared to $\mathcal{O}(N^3)$0 s for IPM and $\mathcal{O}(N^3)$1 ms for numerical bisection. The formula demonstrates speedup factors exceeding $\mathcal{O}(N^3)$2 over standard solvers.

Figure 4: Computation time and speedup ratio.

Dynamic Tracking: Closed-Loop Consistency

End-to-end Monte Carlo experiments with dynamic EKF-based tracking (80 timesteps, $\mathcal{O}(N^3)$3, $\mathcal{O}(N^3)$4 cycles) confirm that the offline performance translates to sustained filtering accuracy and information-theoretic lower bounds on long time horizons, with no filter divergence or error accumulation relative to the IPM optimum.

Figure 5: Dynamic BCRLB.

Figure 6: Dynamic RMSE.

Theoretical and Practical Implications

The research validates that data-free, LLM-driven symbolic regression can autonomously discover compact, physically consistent policies that generalize across scenario scales, operate under tight real-time constraints, and achieve near-optimality for challenging engineering resource allocation problems. This represents a departure from conventional DNN-based approaches, which require extensive training data and are susceptible to distribution shifts.

Strong claims established in this work:

• Near-optimality: $\mathcal{O}(N^3)$5 average relative performance loss, with no catastrophic outliers.
• Generalization: no error inflation under dramatic increases in scenario scale.
• Computational efficiency: over three orders of magnitude real-time speedup ($\mathcal{O}(N^3)$6 vs. $\mathcal{O}(N^3)$7 per IPM iteration).
• Interpretability: the final formula is analytically tractable, facilitating auditing and hardware deployment.

Future Directions

LLM-guided symbolic search for white-box control and resource optimization is an emerging area with substantial unexplored potential. Prospective developments include:

• Extending the technique to multi-radar or networked sensor settings with more sophisticated physical inter-target coupling.
• Dynamic adaptation of the symbolic models based on online feedback without human intervention.
• Real-time code-generation and deployment of evolved formulas in hardware-constrained, distributed radar networks, pushing the limits of edge intelligence.
• Application to other convex and nonconvex engineering optimization domains beyond radar, such as wireless communications, resource management in satellite constellations, and energy grid optimization.

Conclusion

LLM-augmented symbolic evolution, as exemplified by the AlphaEvolve paradigm, provides a principled path to iteration-free, closed-form resource allocation strategies in demanding engineering systems. The approach achieves near-optimal tracking performance, physics-consistent interpretability, real-time computational suitability, and robust generalization—all without requiring large-scale training data. The framework is poised to drive advances not only in radar resource management but also in a broad spectrum of algorithmic optimization challenges in AI-augmented engineering.