Block Coordinate Descent (BCD) Algorithm

• Block Coordinate Descent (BCD) is an iterative optimization method that sequentially updates blocks of variables to solve high-dimensional, often nonconvex problems.
• BCD leverages techniques like Successive Convex Approximation and Majorization-Minimization to ensure convergence even in complex, constrained settings.
• In secure dual-functional radar-communication systems, BCD is applied to jointly optimize waveform, beamforming, and other design variables under strict QoS and power constraints.

Block Coordinate Descent (BCD) Algorithm

Block Coordinate Descent (BCD) is an iterative optimization technique for solving high-dimensional, often nonconvex problems by sequentially optimizing one subset (block) of variables at a time, holding the remaining variables fixed. This methodology is widely adopted in signal processing, wireless communications, machine learning, and, more recently, dual-functional radar-communication (DFRC) system design due to its computational tractability, modularity, and the ability to leverage problem structure.

1. Formal Definition and General Framework

Let $f(x_1, x_2, ..., x_B)$ be an objective function in $B$ blocks of variables, possibly under additional constraints. BCD approaches the solution to

$$\min_{x_1, x_2, ..., x_B} \ f(x_1, ..., x_B) \qquad \text{s.t.} \quad (x_1, ..., x_B) \in \mathcal{C}$$

by cyclically (or in another prescribed order) updating each $x_b$ via

$$x_b^{(k+1)} = \arg\min_{x_b} f(x_1^{(k+1)}, ..., x_{b-1}^{(k+1)}, x_b, x_{b+1}^{(k)}, ..., x_B^{(k)}), \quad \text{subject to constraints in } x_b.$$

The method proceeds until convergence (e.g., no significant decrease in $f$).

For nonconvex, nonsmooth, or constrained problems, BCD can be combined with various tools such as Majorization-Minimization (MM), fractional programming, penalty/augmented Lagrangian techniques, and Successive Convex Approximation (SCA). Each block update may be solved optimally or approximately, and subproblems often result in convex formulations (e.g., Quadratic Programs (QP), Second-Order Cone Programs (SOCP), Semidefinite Programs (SDP)), based on the block's structure.

2. Convergence and Theoretical Properties

The convergence profile of BCD (to global or local optima, or stationary points) is governed by the following conditions:

• For jointly convex problems, each block update is a convex program, and cyclic BCD converges to a global minimum under mild regularity assumptions.
• For nonconvex but continuously differentiable functions with solution sets that are closed and convex in each block, BCD converges to a stationary point if each block subproblem is solved exactly.
• For nonsmooth or constrained settings, variants such as Proximal BCD or MM are used to ensure subproblem tractability and convergence guarantees.

In large-scale or hybrid-architecture systems (e.g., communication systems with digital and analog beamforming), BCD yields efficient low-complexity solvers, often with linear, superlinear, or sublinear convergence depending on problem curvature and block coupling.

3. BCD in Secure Dual-Functional Radar-Communication System Design

BCD is foundational in advanced DFRC system optimization, particularly for joint waveform, beamformer, receiver filter, RIS phase, and array geometry design under security, power, and quality-of-service constraints.

3.1. Secure DFRC with Movable Antennas and RIS

For RIS-enhanced DFRC with movable antennas, the system model comprises a DFRC BS with $N$ movable TX/RX antennas, an $M$-element RIS, $K$ single-antenna users, and targets with clutter (Yang et al., 13 Feb 2025). The main design goal is to maximize radar SINR subject to communication QoS, power, and hardware constraints.

Let $W$ (TX beamformers), $V$ (RIS phases), $\tilde t$ (TX positions), $\tilde r$ (RX positions), and $u$ (receive filter) be the design variables. The core BCD algorithm partitions the variables into blocks and alternates updates:

1. Auxiliary Variable ($\Lambda$): Closed-form update via quadratic transform to linearize the fractional radar-SINR objective.
2. Beamformer Update ($W$): For fixed $\Lambda, V, \tilde t$, SCA is used to convexify nonconvex constraints and subproblems are solved as QPs.
3. RIS Phase Update ($V$): With $W, \Lambda, \tilde t$ fixed, maximize the worst-user SINR subject to unit-modulus RIS constraint; this is handled via penalty methods and SCA, leading again to a convex QP (solved via CV