Alternate Minimization Scheme

Updated 5 December 2025

Alternate Minimization Scheme is an iterative optimization method that decomposes variables into disjoint blocks and updates them sequentially to solve structured problems.
It features exact, proximal, and linearized variants that ensure monotonic descent and convergence under proper regularity conditions.
Its applications span matrix balancing, expectation-maximization, and tensor scaling, making it pivotal in machine learning, numerical linear algebra, and physics.

An alternating minimization scheme is an iterative algorithmic framework for solving structured optimization problems by decomposing the variables into disjoint blocks and successively minimizing over each block while holding the others fixed. This approach, also called block-coordinate descent or alternating projection in certain settings, is fundamental across convex and nonconvex optimization, numerical linear algebra, machine learning, signal processing, and computational physics. The scheme underpins algorithms such as the Sinkhorn scaling for matrix balancing, expectation–maximization, certain primal–dual solvers, and a wide variety of matrix factorization and low-rank recovery methods.

1. Abstract Form and Core Principles

Let $X$ , $Y$ be sets (e.g., $\mathbb{R}^n$ , convex cones, or matrix spaces) and $\Phi : X \times Y \to \mathbb{R} \cup \{+\infty\}$ a function to be minimized. The basic iteration alternates between updating $x \in X$ and $y \in Y$ :

$x^{k+1} = \arg\min_{x \in X} \Phi(x, y^k)$
$y^{k+1} = \arg\min_{y \in Y} \Phi(x^{k+1}, y)$

This iteration, possibly with inexact solves or additional regularization, generates a sequence of iterates $(x^k, y^k)$ which can be shown to converge or contract under appropriate structural and regularity assumptions (Byrne et al., 2015, Sun et al., 2016, Ha et al., 2017).

The alternating scheme can also be generalized to more than two blocks and to include gradient, proximal, or Bregman steps in each coordinate, yielding a large algorithmic family including block-wise coordinate descent, PALM, Gauss–Seidel, and projected minimization (Sun et al., 2016, Zhang et al., 2014, Tupitsa et al., 2019).

2. Algorithmic Templates and Variants

Exact Alternating Minimization

This is the classic two-block case:

$\begin{aligned} x^{k+1} &= \arg\min_x \Phi(x, y^k) \ y^{k+1} &= \arg\min_y \Phi(x^{k+1}, y) \end{aligned}$

This procedure is guaranteed to monotonically decrease $\Phi$ if each subproblem is solved exactly and $X$ , $Y$ are appropriately regular (Byrne et al., 2015).

Proximal and Linearized Variants

To address settings where subproblems are difficult, alternating minimization is often combined with proximal or majorization steps. For instance:

Proximal Alternating Linearized Minimization (PALM): linearizes only the nonconvex part plus adds a quadratic proximal term for tractability (Zhang et al., 2014, Sun et al., 2016).
Bregman-frame AM: employs Bregman divergences as blockwise regularizers and unifies exact, proximal, and linearized AM-type methods in a single template (Sun et al., 2016).

Inexact and Block-Parallel Schemes

In high-dimensional or distributed contexts, subproblems may be solved approximately or in parallel with block-wise updates (Ha et al., 2017, Zhang et al., 2014, Vaswani, 20 Apr 2025, Song et al., 2023).

General Algorithmic Table

Variant	Block update type	Step structure
Exact AM	Exact minimization	Solve each block exactly
Proximal AM	Proximal minimization	Add $\ell_2$ or Bregman penalty
Linearized AM	Prox/linear step	Linearize and proximal/gradient step
Inexact AM	Approximate solve	Controlled subproblem error
Parallel AM	Simultaneous/generic	Update blocks in parallel

3. Equivalence with Majorization–Minimization and Proximal Algorithms

Alternating minimization is equivalent to:

Proximal Minimization Algorithms (PMA): $x^{k+1} = \arg\min_x [f(x) + d(x,x^k)]$ for a suitable distance $d$ induced by the structure of $\Phi$ , where the second block ( $y$ ) minimization defines $f(x)$ and $d(x,x^k)$ (Byrne et al., 2015).
Majorization-Minimization (MM): Alternating minimization steps can be viewed as minimizing a majorizing surrogate that tangentially touches the true objective at the previous iterate.

This foundational equivalence enables transfer of convergence theory—such as monotonic descent, limit identification, and choice of minimizer in the case of non-unique solutions—across frameworks, and relates AM to established proof techniques in convex optimization (Byrne et al., 2015).

4. Convergence Theory: Monotonicity, Rates, and Regularity

Conditions for Convergence

Three-Point Property (3PP) / Weak Three-Point Property (w3PP): If successive minimization steps generate sufficient surrogate decrease, monotonic convergence to a global minimum (in the convex case) or a critical point (in the nonconvex case) follows (Byrne et al., 2015, Sun et al., 2016).
Polyak–Łojasiewicz or Strong Convexity: Linear convergence rates can be achieved under PL conditions or strong convexity, even in block-coordinate or multi-block settings (Tupitsa et al., 2019).
Kurdyka–Łojasiewicz (KL) Property: Establishes global convergence (possibly with a quantified rate) to critical points for nonconvex problems under mild stratified-analytic regularity (Sun et al., 2016).
Sublinear $O(1/k)$ Rates: For convex blocks and sufficiently smooth cross-terms, the objective residual decays as $O(1/k)$ without strong convexity (Zhang et al., 2014).

Inexact and Block-Parallel Schemes

Inexact solutions per block only slow contraction by an additive tolerance propagating through the iteration, but do not fundamentally affect stability or rate if error decays sufficiently fast (Ha et al., 2017, Song et al., 2023).
Parallel updates and block-ordering strategies can preserve $O(1/k)$ rates in convex cases (Zhang et al., 2014).

5. Notable Applications and Theoretical Insights

Sinkhorn Scaling and Matrix Balancing

The Sinkhorn–Knopp algorithm alternates row and column normalization to scale a positive matrix to doubly stochastic form. Each alternation is a Bregman projection onto one constraint set, and strict convexity guarantees global convergence (Nathanson, 2018). This setting is prototypical for understanding alternating minimization as alternating projection in information geometry.

Invariant Theory and Tensor Scaling

The alternating minimization principle generalizes to nonconvex, group-action settings such as tensor scaling, operator scaling, and the null-cone problem in invariant theory. Here, block minimizations are over invertible matrix actions that admit explicit solutions in each coordinate, and a potential-function argument establishes polynomial-time convergence (Bürgisser et al., 2017).

Statistical Learning and Regression

Alternating minimization is fundamental in matrix factorization, weighted low-rank approximation, and multi-task regression. Approximating each factor block by regression or sketch-based solvers retains robust convergence under incoherence and spectral gap assumptions, with runtime improvements via high-accuracy iterative solvers (Song et al., 2023, Vaswani, 20 Apr 2025, Burri, 7 Jul 2025).

Nonconvex and Rate-Independent Evolution

In nonconvex phase-field models or rate-independent variational evolutions, alternate minimization (often called “staggered schemes”) allows for tractable updates in physics-informed models and lends itself to rigorous analysis of convergence to viscous or balanced-viscosity solutions (Boddin et al., 2022, Almi, 2019).

Minimax and Adversarial Optimization

Alternated gradient-descent–ascent (Alt-GDA) steps in minimax settings (e.g., GANs, saddle-point problems) achieve improved iteration-complexity and linear convergence compared to simultaneous updates, bridging to extra-gradient methods with optimal rates in convex–concave games (Lee et al., 16 Feb 2024).

6. Advanced Topics and Recent Extensions

Bregman-Frame AM: Unifies and interpolates between pure, proximal, and linearized schemes, showing that strong convexity and the KL property control global rates and critical point convergence (Sun et al., 2016, Zhang et al., 2014).
Higher-Order and Parallel Block Extensions: Generalization to more than two blocks, cyclic or greedy block selection, and parallel architectures (Tupitsa et al., 2019, Vaswani, 20 Apr 2025).
Federated and Decoupled Optimization: Alternating gradient-descent–minimization (AltGDmin) achieves reduced computation and communication in federated learning and large-scale distributed systems where one block is decoupled and cheap to minimize (Vaswani, 20 Apr 2025).
Regularization and Smoothing Approaches: Prox-function smoothing, Fenchel-type operators, and Nesterov acceleration in alternating minimization lead to optimal $O(1/\epsilon)$ or $O(1/\epsilon^2)$ complexity rates for primal-feasibility and residual, even without full strong convexity (Tran-Dinh, 2015).

7. Theoretical and Practical Insights

Alternating minimization methods form a ubiquitous and flexible algorithmic toolkit, with the following key attributes:

Provable Monotonicity: Every method in the class ensures a nonincreasing objective.
Flexibility in Regularity: Methods remain robust under nonsmoothness, nonconvexity, and blockwise separability assumptions, especially when combined with inexact updates.
Equivalence and Unification: AM, PMA, and MM are mathematically equivalent, yielding transferability of proof strategies and rate results (Byrne et al., 2015).
Extension to Complex Domains: The framework naturally extends to manifold constraints (e.g., quantum density matrices), group actions, and non-Euclidean geometries.
Efficient Implementation: Alternating minimization is particularly appealing where each block-update is closed-form, low-dimensional, or parallelizable—resulting in runtime and communication advantages in modern large-scale applications (Song et al., 2023, Vaswani, 20 Apr 2025).
Convergence Rates and Limitations: Linear convergence is attainable under strong convexity or PL conditions, but for many nonconvex or non-smooth models only sublinear rates or local convergence can be guaranteed. Adaptive selection of block ordering or regularizers to speed convergence remains an open challenge (Sun et al., 2016).