Alternating Optimization Algorithm

• Alternating Optimization (AO) is a framework that decomposes decision variables into blocks, optimizing one block at a time while keeping others fixed.
• AO enables scalable solutions for high-dimensional, nonconvex, and constrained problems by ensuring convergence under specific regularity and local geometric conditions.
• Variants such as proximal-gradient, stochastic, and hybrid AO extend its applicability to matrix factorization, regression, and wireless communication challenges.

Alternating Optimization (AO) is a foundational framework for solving nonconvex and/or constrained optimization problems in which the decision variables naturally decompose into distinct blocks. AO exploits this partitioning by sequentially optimizing over each block while the others are held fixed. This modular strategy enables scalable algorithms for high-dimensional and structured problems across applied mathematics, statistics, signal processing, and communications. AO admits convergence guarantees under a range of regularity and problem-specific conditions, supports efficient handling of hard and soft constraints, and forms the theoretical basis for a broad class of stochastic, inexact, and extended algorithms.

1. Core Principles and Problem Formulation

Let the variables be partitioned as $\{x_b\}_{b=1}^B$, with each $x_b$ constrained to a (possibly nonconvex) set $X_b$, and let $f(x_1,\ldots,x_B)$ be a differentiable objective (possibly nonconvex with respect to the joint variables). The canonical AO method proceeds iteratively:

• At iteration $t$, for block $b$, compute

$$x_b^{(t+1)} = \arg\min_{x_b \in X_b} f(x_1^{(t+1)}, \ldots, x_{b-1}^{(t+1)}, x_b, x_{b+1}^{(t)}, \ldots, x_B^{(t)}).$$

Under reasonable continuity and bounded-level assumptions, the AO sequence $\{f(x_1^{(t)}, \ldots, x_B^{(t)})\}$ is nonincreasing, and any limit point is blockwise optimal (i.e., stationary with respect to AO steps) (Murdoch et al., 2014).

For constrained/decomposable objectives with a smooth loss plus separable regularizers, e.g.,

$$\min_{x\in X,\, y\in Y} F(x,y) = f(x) + g(y) + H(x,y),$$

AO typically alternates between proximal-gradient-based or variational subproblem solutions for each block (Driggs et al., 2020).

2. Theory: Convergence, Local Concavity, and Conditioning

For nonconvex, block-structured problems, AO's convergence hinges on blockwise tractability, local geometric regularity, and initialization:

• Local Concavity Coefficient $\gamma_z(Z)$: For a (possibly nonconvex) feasible set $Z$, its “curvature condition” at $z \in Z$ measures the deviation from convexity, determining the strength of necessary optimality conditions for AO (Ha et al., 2017). If $Z$ is convex, then $\gamma_z(Z)=0$ and standard KKT theory applies.
• Restricted Strong Convexity (RSC) / Smoothness (RSM): If the objective is (locally) strongly convex and smooth with respect to each block (possibly within a restricted set induced by initializations), then AO can contract distances to a global or local minimizer at a linear rate, modulated by the local concavity and the condition numbers of the block Hessians (Ha et al., 2017).
• Inexact AO: Practical AO often uses iterative, approximate block updates (e.g., projected gradient descent, primal-dual splitting as in AO-PDS (Ono et al., 2017)). Provided subproblem errors decay appropriately (proportional to block iterates), convergence results from the exact AO extend, typically with only logarithmic impact on outer iteration counts.
• Comparison to Joint Gradient Descent: AO may substantially outperform joint gradient descent when the condition numbers of the blockwise subproblems are imbalanced, since AO’s rate is determined by the better (smaller) blockwise condition number (Ha et al., 2017).

3. Algorithmic Extensions and Variants

Several important extensions of the generic AO paradigm have been developed to improve convergence, robustness, and scalability:

• Expanded AO: After AO converges to a coordinate-wise (blockwise) stationary point, an “escape” phase performs additional optimization steps in low-dimensional but problem-informed subspaces $S_\text{escape} \supset S_\text{AO}$ (e.g., joint updates over correlated block subsets or principal directions), empirically helping to avoid saddle points and poor local minima (Murdoch et al., 2014).
• Stochastic and Variance-Reduced AO: For large-scale finite-sum (data-fitting) problems, stochastic block updates with variance reduction (e.g., SAGA, SARAH) are applied to the smooth coupling term in each block, reducing per-iteration cost and restoring deterministic convergence rates in expectation. For instance, the SPRING algorithm achieves $O(1/k)$ suboptimality decay and linear rates under error-bound conditions in nonconvex optimization (Driggs et al., 2020).
• Alternating Optimization for Bi-Objective Problems: Stochastic alternating approaches (SA2GD/SA2SG) solve scalarized multi-objective functions by alternating blocks of gradient/subgradient steps on each constituent objective, enabling approximation of the Pareto front across weighted combinations (Liu et al., 2022).
• Hybrid AO-Plus Subspace or Proximal Steps: Practical schemes often replace exact block solves with a few sub-iterations (proximal-gradient, primal-dual splitting (Ono et al., 2017)), which significantly accelerates convergence for composite-structured constraints (e.g., nonnegativity, sparsity, total variation).

The following table outlines representative AO variants and their key design features:

Variant	Subproblem Solution	Scalability/Advantage
Classical AO	Global block minimization	High accuracy when subproblems easy
Proximal-Gradient AO	Linearize + prox step	Closed-form, low per-iteration cost
Stochastic Variance Reduction	SAGA/SARAH blocks	Scalable on large finite-sums
Expanded AO	Joint/subspace search	Escapes blockwise-stationary points
Primal-Dual Splitting (AO-PDS)	Dual ascent on composites	Handles hard+soft constraints flexibly

4. Applications and Domain-Specific AO Frameworks

AO is widely adopted in numerous large-scale and structured statistical and signal processing problems:

• Matrix and Tensor Factorization: AO provides the backbone for canonical polyadic tensor decomposition with hard constraints (nonnegativity) and soft structure (e.g., group sparsity). The AO-PDS scheme splits subproblems into convex composite programs, solving each via matrix-inversion-free primal-dual splitting, offering significant improvements over classical AO-ADMM both in runtime and flexibility (Ono et al., 2017).
• Robust Principal Component Analysis and Multi-Task Regression: AO alternates over low-rank and sparse matrix components (for decomposition) or over regression matrix and noise covariance, with local concavity coefficients providing theoretical control over convergence rates and accuracy, including the impact of initialization and regularization (Ha et al., 2017).
• Penalized Regression and Recommender Systems: AO underlies coordinate-descent in high-dimensional regression with structured sparsity (e.g., MC+ penalty), and alternating least-squares in collaborative filtering. Expanded AO, accessing correlated covariates or principal directions in the joint parameter space, demonstrably improves empirical risk and variable-selection objective (Murdoch et al., 2014).
• Wireless Communications: In sum-rate or SNR maximization under nonconvex coupled constraints (IRS/beamformer phase shifts, meta-surface design), AO alternates efficient updates on phase and beamformer variables, sometimes with successive convex approximations. The design order (e.g., updating meta-surface phase before digital beamformer) and the inner-loop schedule of prox-based updates have been shown to dramatically influence the achievable rate and convergence speed (Bahingayi et al., 21 Aug 2025, Zhou et al., 28 Apr 2025). In some tractable channel configurations, the AO loop can even be avoided altogether, with closed-form decoupled solutions yielding identical performance at lower computational cost (Hu et al., 7 May 2024).

5. Advanced Analysis: AO in Nonconvex and Structured Settings

The reach and limitations of AO in highly-structured, nonconvex, and constrained settings have been rigorously analyzed:

• Convergence Guarantees: Under joint restricted strong convexity, restricted smoothness, controlled cross-block gradient coupling, and suitable initialization, AO achieves provable linear contraction to a local minimizer. The key novelty lies in formalizing nonconvex set geometry via local concavity coefficients, which capture the loss of first-order optimality at nonconvex points (Ha et al., 2017).
• Role of Block Conditioning: AO exploits separability: its convergence is dictated by the smallest blockwise condition number, in contrast to joint optimization methods, which are limited by the worst case. This translates to marked empirical advantages when problem structure yields favorable blockwise geometry.
• Inexact Subblock Optimization: Provided subblock optimization error is proportional to the current iterate's step size, overall AO convergence is robust to imperfect block solves, as evidenced by both theory and practice in high-dimensional tensor models and regularized nonnegative factorizations (Ono et al., 2017).
• Identifying When AO Is Unnecessary: If, at the problem level, the dependence of the global objective on each block reduces to a monotonic function of a single block-specific surrogate (e.g., channel correlation in multicasting), AO may be superfluous; closed-form sequential optimization is provably as powerful, leading to both computational and theoretical transparency (Hu et al., 7 May 2024).

6. Practical Guidelines and Empirical Observations

• Initialization: Strong initializations are crucial for fast local convergence and avoidance of spurious stationary points, particularly in nonconvex landscapes with local concavity.
• Update Order: In certain multi-block applications (e.g., SIM-based multiuser MISO systems), updating blocks with more impactful degrees of freedom first yields significantly better performance and faster convergence, as observed when phase shifts are updated before digital beamformers (Bahingayi et al., 21 Aug 2025).
• Blockwise Subspace Selection: Choosing search directions informed by residual gradients, principal directions, or highly correlated variable subsets can accelerate convergence and escape shallow local minima (Murdoch et al., 2014).
• Algorithmic Tuning: Step-size selection, stopping criteria (with separate thresholds for AO versus subspace/escape phases), and inner-iteration schedules (for PG or PDS solvers) should be tuned according to local data geometry and empirical validation.
• Computational Complexity: The cost per AO iteration is often determined by the block dimension and complexity of prox or projection operators. For high-rank tensors, primal-dual splitting removes the need for cubic scaling matrix inversions, enabling much faster convergence per outer iteration (Ono et al., 2017).
• Stochastic AO: For large data or streaming, stochastic block updates, with proper memory scheduling and variance reduction, make AO methods robust and scalable while retaining convergence guarantees (Driggs et al., 2020). For bi-objective optimization, alternating the number of steps spent on each component enables traversal of the Pareto frontier with only sublinear convergence rate decay (Liu et al., 2022).

In aggregate, these principles render AO a flexible, theoretically grounded, and highly practical meta-approach for a diverse range of block-structured and nonconvex optimization problems. For comprehensive mathematical details and advanced algorithmic implementations, see (Ha et al., 2017, Murdoch et al., 2014, Ono et al., 2017, Driggs et al., 2020, Hu et al., 7 May 2024, Bahingayi et al., 21 Aug 2025, Zhou et al., 28 Apr 2025), and (Liu et al., 2022).