Part 2: Beamforming Optimization

Chapter 6: Phase Optimization Algorithms

Advanced~230 min

Learning Objectives

  • Derive the SDR relaxation of the unit-modulus quadratic and perform Gaussian randomization
  • Understand manifold optimization on the complex torus: Riemannian gradient and retraction
  • Implement element-wise BCD and establish its monotone convergence
  • Compare SDR, manifold, and element-wise algorithms on solution quality vs. runtime
  • Choose the right algorithm for a given deployment scenario

Sections

Prerequisites & Notation
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The Unit-Modulus QCQP
Semidefinite Relaxation and Gaussian Randomization
Manifold Optimization on the Complex Torus
Element-wise Block Coordinate Descent
Comparing SDR, Manifold, and Element-wise
Chapter Summary
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Exercises
References & Further Reading
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Prerequisites

Chapter 5: passive subproblem formulationSemidefinite programming, SDR relaxation (Telecom Ch. 3)Riemannian geometry: tangent space, retraction (optional; developed here)Gaussian distributions on the complex unit sphereKKT and first-order optimality

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