Worst-Case and Covariance-Reconstruction Robust Adaptive Beamforming: Equivalence and Closed-Form Solution

Robust adaptive beamforming (RAB) has developed along two separate lines. The worst-case-constrained approach guards against steering vector mismatch with a bounded uncertainty set, but suffers from desired signal cancellation at high input signal-to-noise ratio (SNR). The covariance-reconstruction approach instead estimates the interference-plus-noise covariance matrix and solves for the mismatch numerically. We prove that the two are related by primal-dual equivalence: once the worst-case beamformer is normalized, it coincides exactly with the reconstruction-based one, so both reduce to one formulation. The signal cancellation of the worst-case approach is a consequence of its parameter choice, not of the formulation. For the unified problem we develop a closed-form solution in three stages: Diagonalization Transform, Phase Alignment, and Karush-Kuhn-Tucker (KKT) Solution. Instead of solving for the Lagrange multiplier, we introduce an auxiliary variable that reduces the KKT system to a single scalar equation solvable by bisection. For an $N$-sensor array the cost is $\mathcal{O}(N^{3})$, the same order as the Lagrange-equation-based RMVB and well below the $\mathcal{O}(N^{3.5})$ of a second-order cone program, and it covers the rank-deficient case that RMVB excludes. We establish necessary and sufficient conditions for existence and uniqueness, including a case where the feasible set is nonempty yet no optimal solution exists, and show that the Lagrange multiplier fixes the optimal value. Simulations confirm that, with matching parameters, the worst-case beamformer attains the output signal-to-interference-plus-noise ratio (SINR) of the reconstruction-based one, and that the closed-form solution reproduces the optimum of convex solvers in a fraction of their time.