Block Diagonal Relaxations in Optimization

• Block diagonal relaxations are techniques that approximate full matrix constraints using block structures to simplify large-scale optimization and control problems.
• They enable parallelizable and scalable algorithms by decomposing matrices into independent subproblems, as seen in preconditioning and Lyapunov analysis.
• Applications span convex programming, semidefinite optimization, and deep neural network training, balancing reduced computational cost with potential conservatism.

Block diagonal relaxations are a class of techniques in numerical optimization, matrix analysis, control theory, and large-scale computation that exploit block structure in matrices to decompose, simplify, or approximate challenging problems. Such methods encompass the replacement of full (possibly dense) matrices or constraints with relaxed (block-diagonal or block-structured) forms. This allows leveraging parallel architectures, reducing computational burdens, and obtaining scalable algorithms without severely degrading problem fidelity. Block diagonal relaxations pervade contexts from preconditioning for linear system solvers, relaxation of convex programs, scalable Lyapunov analysis, to block-structured training in deep neural networks.

1. Definition and General Principle

Block diagonal relaxation is the process of substituting a full matrix, operator, or constraint with its block-diagonal part or a block-diagonal approximation. For a matrix $A\in\mathbb{R}^{N\times N}$, partitioned into blocks $A_{ij}$, its block-diagonal is $\text{diag}(A_{11},...,A_{nn})$. In optimization and numerical linear algebra, this relaxation involves solving or optimizing over this reduced structure instead of the original, typically denser, formulation.

Block diagonalization is exploited in:

• Preconditioning, where block-diagonal preconditioners approximate more complex matrices,
• Convex relaxations, as in conic or semidefinite programming (SDP) by replacing difficult constraints with block-diagonal LMIs,
• Stability and control via block-diagonal Lyapunov certificates,
• Parallelizable iterative solvers using block-wise updates.

This principle often yields methods with independent subproblem structure, amenable to distributed and parallel computing, and scalable to very large system sizes.

2. Block Diagonal Relaxations in Numerical Linear Algebra

Block Relaxation for Linear Systems

Block-Jacobi and block Gauss–Seidel methods partition the variable vector according to blocks, then at each iteration update each block by inverting only its associated diagonal block. For example, consider a linear system $A u = f$ with $A$ partitioned as $[A_{ij}]$. The block Jacobi iteration takes

$u^{k+1} = u^k + M^{-1}(f - A u^k),$

where $M = \text{diag}(A_{11},...,A_{ss})$. When each block is small, the cost of inverting each block is amortized over many iterations or re-used across sweeps. Block-Jacobi easily parallelizes, as updates to blocks are independent. In practice, block sizes are chosen as geometric subdomains (e.g., $8^3$ blocks for 3D elliptic stencils), and exact block inverses are precomputed. Multicore and GPU implementations using these methods achieve high parallel efficiency and strong scaling (Birke et al., 2012).

Block Diagonal Preconditioning

For saddle-point and more general $2 \times 2$ block systems,

$$A = \begin{pmatrix} A_{11} & A_{12} \ A_{21} & A_{22} \end{pmatrix},$$

block-diagonal preconditioners $M_{BD} = \text{diag}(A_{11}, S)$ (with $S$ the Schur complement) often yield competitive convergence for Krylov subspace iterations like GMRES or MINRES. However, for non-saddle-point problems ($A_{22} \neq 0$), block-diagonal preconditioning does not guarantee an $O(1)$ convergence rate, in contrast to block-triangular or LDU preconditioning. Nevertheless, block-diagonal approaches may be preferable when the cost of off-diagonal blocks is prohibitive or when inverting full blocks is challenging (Southworth et al., 2020).

3. Block Diagonal Relaxations in Convex Optimization and SDPs

Scalable Lyapunov Analysis via Block-Scaled Diagonal Dominance

Stability of large-scale dynamical systems is often certified by finding positive definite $P$ satisfying the continuous Lyapunov inequality $PA + A^T P \prec 0$. Classical diagonal dominance results guarantee existence of diagonal $P$ under certain conditions. Block diagonal relaxation generalizes this: if a block partitioned matrix $A$ is block-scaled diagonally dominant (BSDD), then a block-diagonal $P$ exists, satisfying partial Lyapunov inequalities for each block. This enables replacing a large $N \times N$ LMI with $n$ smaller $k_i \times k_i$ LMIs/Riccati inequalities, drastically reducing computational complexity and enabling parallel/distributed solutions, at the cost of some conservatism (Sootla et al., 2017).

Block Diagonal Relaxations in Semidefinite Programming

SDPs with block-diagonal constraints, such as

$$\min_{X\succeq0}\ f(X) \quad \text{s.t.} \ X_{ii}=B_i,\, X_{ij}=0 \text{ for } i\neq j,$$

appear in synchronization, combinatorial optimization (e.g., Max-Cut), and pose graph estimation. Block-diagonal relaxation replaces global PSD or LMI constraints with block-wise versions. The Burer–Monteiro factorization leverages this by optimizing over block-factorized variables embedded in a product of Stiefel manifolds. Block-coordinate minimization (BCM) and Riemannian staircase methods allow scalable, memory-efficient algorithms with provable global optimality for rank-deficient accumulations (Tian et al., 2019, Boumal, 2015).

For graph-theoretic SDPs, block-diagonal relaxation also exploits symmetry to partition the main SDP block into smaller orthogonal blocks, dramatically reducing computational cost for problems like the clique number of Paley graphs, at some expense of tightness compared to full SOS relaxations (Kobzar et al., 2023).

Strengthened Block Diagonal SDP Relaxations via Structure

For quadratic optimization over the Stiefel manifold, block-diagonal Hessian structure can be leveraged to enforce diagonal-sum LMIs ($\sum_j X_{jj} \preceq I_n$). This stronger relaxation tightens the feasible set. The Kronecker-product construction adds further constraints, yielding improved performance, especially for random or structured problem instances, at the expense of much higher per-iteration complexity and larger semidefinite constraints (Burer et al., 2022).

4. Block Diagonal Approximations in Machine Learning

Block-diagonal relaxations are directly applied to large-scale neural network optimization. In Hessian-free training, the full generalized Gauss–Newton matrix $G$ is replaced by its block-diagonal part, where parameters are grouped by layer, time step, or architectural blocks. Each block-wise quadratic subproblem is solved independently by conjugate gradient. This preserves within-block curvature while discarding cross-block curvature, resulting in lower computational cost, improved parallelism, and reduced sensitivity to noise in large mini-batch or curvature-batch regimes. Empirical results show convergence speed, robustness, and scaling improvements relative to both first-order (Adam/SGD) and full HF second-order methods, particularly when parallel resources are exploited (Zhang et al., 2017).

5. Adaptive and Inexact Block Diagonal Relaxation Strategies

Relaxation strategies in iterative solvers using block-diagonal structure include dynamically adapting solution tolerances for inner block systems, based on outer iteration coefficients. For saddle-point systems solved by iterative Krylov subspace or inner-outer Golub–Kahan bidiagonalization (GKB), the residual coefficients $\zeta_k$ decay superlinearly. Setting inner solver tolerance $\tau_k \propto 1/|\zeta_{k-1}|$ or switching to a predicted form enables significant savings in total inner iterations without sacrificing global solution accuracy. Hybrid selection among several candidate tolerances further minimizes cost. These methods yield up to 50–60% reduction in computational effort versus fixed-tolerance schemes, and generalize to broader classes of block systems beyond classical PDE examples (Darrigrand et al., 2022).

6. Block Diagonal Relaxations in Data Analysis and Subspace Clustering

Explicit block-diagonal regularization is used to recover coefficient matrices with block structure, underpinning methods such as block-diagonal representation (BDR) and adaptive block-diagonal representation (ABDR) for subspace clustering. ABDR uses a convex bi-fusion regularizer that forces both columns and rows of the coefficient matrix to fuse adaptively, guaranteeing block-diagonal structure under mild conditions. This approach unites the convexity and parameter simplicity of "indirect" priors (sparsity/low-rankness) with the block-structure enforcement of "direct" regularization, achieving globally optimal solutions and strong empirical performance on benchmark clustering problems (Lin et al., 2020).

7. Open Problems, Limitations, and Trade-offs

Block diagonal relaxations offer parallelism, scalability, and reduced computational complexity but have inherent limitations:

• The neglect of cross-block couplings can lead to conservatism in Lyapunov and SDP relaxations or slower convergence in optimization.
• The effectiveness of block-diagonal preconditioners depends on the spectral properties of the problem (e.g., whether the off-diagonal blocks are small or the system is near-saddle-point structure).
• For non-saddle-point block-linear systems, block-diagonal preconditioning does not yield uniformly fast convergence (Southworth et al., 2020).
• In convex relaxations, the relaxation gap increases when structure is disregarded; e.g., full Kronecker-product constraints or higher level SOS relaxations close more of the feasible region but increase computational cost (Burer et al., 2022).
• While block coordinate algorithms scale well, global optimality is only guaranteed for rank-deficient critical points or when specific nontrivial geometric properties hold (Tian et al., 2019, Boumal, 2015).

Overall, the deployment of block-diagonal relaxation is governed by balancing fidelity against computational tractability, with application-specific considerations dictating the optimal relaxation scheme and block configuration.