Surrogate-Assisted Block-Coordinate Search

• The paper demonstrates that surrogate-assisted block-coordinate search efficiently decomposes high-dimensional, non-smooth problems, drastically reducing computational cost while achieving near-optimal solutions.
• This approach partitions complex optimization tasks into blocks, applying surrogate models to approximate expensive evaluations, which is crucial in radio-frequency and wireless antenna designs.
• The methodology integrates analytical relaxations and regression surrogates, validated through assignment algorithms and reinforcement learning, to navigate discontinuous objective landscapes.

Surrogate-assisted block-coordinate search is an optimization paradigm in which global or large-scale mixed-variable problems are decomposed into blocks, each block is optimized conditionally on others, and expensive subproblems are approximated using surrogates to expedite search. This approach is particularly relevant when evaluating the true objective for certain variables is computationally expensive, non-smooth, or discontinuous, as is common in radio-frequency and wireless system design involving physical placement, resource allocation, and channel-dependent performance metrics. Surrogate models—analytical approximations or data-driven regressors—replace direct evaluation during intermediate search, reducing computational cost while seeking near-optimal block-wise solutions.

1. Block-Coordinate Decomposition and Surrogate Integration

Block-coordinate search partitions the optimization variables into disjoint blocks. At any iteration, one block is optimized while others are held fixed, often alternating across all blocks until convergence. For each block, direct objective evaluation may be prohibitive, especially if it involves Monte Carlo simulations, solution of implicit equations, or run-time resource allocation. Surrogate modeling addresses this by replacing the true block-wise objective with a cheaper-to-evaluate approximation, typically trained on prior evaluations or employing analytical relaxations. The surrogate is updated alongside the search, either via offline pre-training or on-the-fly augmentation.

In indoor pinching-antenna systems, discrete PA (pinching-antenna) position optimization—the task of selecting deployment positions along a waveguide under physical and LoS (line-of-sight) constraints—was addressed as a special case using surrogate-assisted block-coordinate search. The task decouples the clustered combinatorial assignment (e.g., via the Hungarian algorithm) from subsequent local refinement of PA positions conditioned on assignments, wherein a surrogate is used to estimate refined placement performance efficiently (Xie et al., 3 Jan 2026).

2. Application to Pinching-Antenna Systems in Obstacle-Rich Environments

Pinching-antenna architectures, allowing dynamic physical placement of flexible antennas, are sensitive to obstacle-induced LoS blockage. In environments with complex obstacle geometries, the LoS indicator map, $\alpha_{k,m}(\Psi)$, is a binary-valued and discontinuous function of the PA positions $\Psi$. The sum-rate optimization problem—maximizing achievable rates under power and QoS constraints—thus becomes non-smooth with respect to $\Psi$.

For the special case where each PA is discretely deployed and serves a single user, the waveguide-user assignment is computed via the Hungarian algorithm based on current LoS and blockage states. Subsequent discrete PA positions are further refined using a surrogate-assisted block-coordinate search, wherein a surrogate objective models the system throughput or user rate for candidate positions, providing fast feedback for local search moves (Xie et al., 3 Jan 2026). This facilitates efficient navigation of the rugged and high-dimensional position space affected by discrete LoS transitions.

3. Surrogate Model Construction and Selection

Surrogate models in block-coordinate search are chosen according to the problem structure and computational tractability. Common surrogates include:

• Analytical relaxations: Simplified deterministic models as proxies for exact physics or combinatorial cost.
• Regression/statistical surrogates: Interpolants or regressors (e.g., Gaussian process, random forest) fit to previously evaluated positions and their performance metrics.
• Piecewise approximations: Structured surrogates precisely capturing boundary-induced discontinuities, as in LoS transitions due to obstacles.

In the context of blockage-aware PA systems, the surrogate may combine a deterministic geometric LoS model with a fast analytic expression for rate or SNR as a function of position, updating as more evaluations accumulate or as the environment changes.

4. Algorithmic Workflow and Block-Coordinate Search Steps

The surrogate-assisted block-coordinate search operates in alternation:

1. Assignment step: Solve the PA-user assignment problem, fixing either PA or user positions (e.g., via Hungarian algorithm for discrete assignments).
2. Position refinement step: For each PA (or block of PAs), optimize the local objective proxy (via surrogate model) with all other settings fixed.
3. Update cycle: Repeat block-wise optimization until performance plateaus or a pre-set iteration bound is met.

This process leverages surrogates to limit calls to the high-fidelity but slow objective—typically reserved for final verification or selective updates. The approach is especially effective when performance metrics exhibit sharp, structured discontinuities (such as LoS boundary crossings), which are computationally expensive to resolve via brute force or grid search.

5. Advantages, Limitations, and Performance Insights

Surrogate-assisted block-coordinate search enables tractable optimization over high-dimensional, combinatorial, and non-smooth design spaces when traditional global search is impractical. In "Pinching Antennas in Blockage-Aware Environments: Modeling, Design, and Optimization," this methodology yielded substantial improvements in throughput and connectivity under realistic obstacle configurations, outperforming grid search and random placement baselines for discrete PA deployment (Xie et al., 3 Jan 2026).

A plausible implication is that the modular decomposition via block-coordinate search aligns well with the inherent separability of physical assignment and fine-grained movement in wireless antenna design, and surrogates mitigate the non-smoothness caused by obstacle-induced LoS shifts.

6. Connections to Joint Optimization and Reinforcement Learning

For continuous PA positioning and joint beamforming, the block-coordinate framework is extended by integrating closed-form weighted-MMSE beamforming (for continuous variables) with deep reinforcement learning (specifically, DDPG-based actor-critic) for global PA placement, as direct joint optimization over position and waveform is not viable for non-smooth LoS maps. In this extension, the block-coordinate spirit is preserved by alternating between solving for fast beamforming (via WMMSE) and global PA placement (via RL), with surrogate penalty functions smoothing over discontinuities. The surrogate-assisted discrete search thus forms a conceptual bridge between pure combinatorial assignment and gradient-based, continuous-space policy learning (Xie et al., 3 Jan 2026).