An Inner SOCP Approximate Algorithm for Robust Adaptive Beamforming for General-Rank Signal Model

The worst-case robust adaptive beamforming problem for general-rank signal model is considered. Its formulation is to maximize the worst-case signal-to-interference-plus-noise ratio (SINR), incorporating a positive semidefinite constraint on the actual covariance matrix of the desired signal. In the literature, semidefinite program (SDP) techniques, together with others, have been applied to approximately solve this problem. Herein an inner second-order cone program (SOCP) approximate algorithm is proposed to solve it. In particular, a sequence of SOCPs are constructed and solved, while the SOCPs have the nondecreasing optimal values and converge to a locally optimal value (it is in fact a globally optimal value through our extensive simulations). As a result, our algorithm does not use computationally heavy SDP relaxation technique. To validate our inner approximation results, simulation examples are presented, and they demonstrate the improved performance of the new robust beamformer in terms of the averaged cpu-time (indicating how fast the algorithms converge) in a high signal-to-noise region.

💡 Research Summary

The paper tackles the worst‑case robust adaptive beamforming (RAB) problem for general‑rank signal models, where the desired signal covariance matrix is only imperfectly known and must satisfy a positive semidefinite (PSD) constraint. Traditional approaches formulate the problem as a semidefinite program (SDP) and solve a sequence of SDPs, which becomes computationally prohibitive as the array size or snapshot number grows.

The authors propose a fundamentally different strategy: an inner second‑order cone program (SOCP) approximation. Starting from the worst‑case SINR maximization formulation, they rewrite the problem (equations (11)–(14)) and isolate a non‑convex quadratic constraint involving the desired signal covariance. By linearizing this constraint around the current iterate (\mathbf w^{(k)}) they obtain a convex SOC constraint that defines a subset of the original feasible set. Consequently, each iteration solves a restricted SOCP (equation (17) or (19)) whose optimal value is a lower bound on the true optimum, but the bound improves monotonically because the feasible set is updated with the newly obtained solution.

Key theoretical properties established in the paper are:

1. Monotonicity – the objective values of successive SOCPs are non‑decreasing (Proposition III.1). This follows from the fact that the current solution always satisfies the next iteration’s linearized constraint.

2. Convergence – because the sequence of lower bounds is bounded above by the true optimum, the algorithm converges. Empirical results show convergence to the globally optimal value, consistent with earlier DC‑POTDC analyses that guarantee global optimality when the covariance error norm is sufficiently small.

3. Complexity reduction – solving an SOCP with interior‑point methods requires roughly (\mathcal O(N^{3.5})) operations, whereas each SDP in the conventional approach costs (\mathcal O(N^{6})). For a 10‑element uniform linear array (ULA) with hundreds of snapshots, the proposed method yields a dramatic reduction in CPU time.

Simulation experiments use a 10‑element ULA, INR = 20 dB, and a range of SNR values (0–60 dB). The desired signal and interferer are modeled as locally incoherently scattered sources with Gaussian angular power densities. The authors compare three quantities: (i) the average CPU time of the new SOCP‑based algorithm, (ii) the CPU time of the K‑V algorithm from