Chapter Summary

Chapter Summary

Key Points

• 1.

  The joint active-passive beamforming problem is non-convex. Maximize $\sum_k \log_2(1 + \text{SINR}_k)$ over $(\mathbf{W}, \boldsymbol{\Phi})$ subject to power constraint and unit-modulus $|\phi_n| = 1$. The unit-modulus torus is non-convex, and the objective is bilinear rather than jointly concave. No polynomial-time global algorithm is known.

• 2.

  But the conditional subproblems are tractable. For fixed $\boldsymbol{\Phi}$, the active subproblem is standard MU-MIMO precoding — convex under WMMSE reformulation. For fixed $\mathbf{W}$, the passive subproblem is a QCQP on the complex torus, solvable by SDR, manifold, or element-wise methods (Chapter 6).

• 3.

  Alternating optimization (AO) is the practical workhorse. Iterate: active update via WMMSE, passive update via Chapter 6 algorithm. Each iteration is monotonically non-decreasing in the objective; convergence to a stationary point is guaranteed under mild regularity conditions. Typical convergence: 10–30 iterations for $\epsilon = 10^{-3}$.

• 4.

  Local vs. global optima. AO produces local optima. With 5-20 random initializations and taking the best, the AO result is typically within $1$-$5\%$ of the SDR upper bound — good enough for nearly all practical purposes. Warm-starting across coherence blocks accelerates real-time operation.

• 5.

  The RIS pattern is just MU-MIMO plus one extra variable. If you already know MU-MIMO precoding, the RIS joint problem is a natural extension: $\mathbf{W}$ still shapes the transmit signal, and $\boldsymbol{\Phi}$ shapes the channel itself. The outer AO loop interleaves these two shaping operations; the theoretical $N^2$ coherent gain is realized algorithmically by alternating between them.