Alternating Optimization Algorithm

• AO-based algorithm is a structured optimization method that decomposes multivariate problems by alternately updating disjoint variable blocks.
• It employs innovations like joint subspace searches, distributed updates, and inexact/stochastic corrections to overcome local minima and improve convergence.
• Applications span signal processing, quantum optimization, machine learning, and resource allocation, offering scalable solutions for complex nonconvex challenges.

Alternating Optimization (AO)–Based Algorithm refers to a general class of structured optimization procedures wherein variables are partitioned into disjoint sets, and the optimization alternates between these sets—systematically updating one subset while keeping the remaining variables fixed. This block-coordinate philosophy leverages problem decomposability and is foundational in high-dimensional, nonconvex, distributed, and constrained optimization scenarios. Substantial applications appear across signal processing, quantum algorithms, machine learning, resource allocation, and control.

1. Foundational Principle and Framework

The core principle underlying AO is the decomposition of a multivariate (often nonconvex) objective into a sequence of conditional subproblems. Let $z = (z_1, \ldots, z_B)$ denote the variable blocks partitioning $\mathbb{R}^{d}$, with each $z_b$ being a (possibly vector-valued) coordinate. AO proceeds by cycling through the blocks, at each iteration solving: $$z_b^{(t)} = \arg\min_{z_b} f(z_1^{(t)}, \ldots, z_{b-1}^{(t)}, z_b, z_{b+1}^{(t-1)}, \ldots, z_B^{(t-1)}),$$ while all other $z_{-b}$ are held at their previous values. The feasible set for each subproblem may be nonconvex or subject to constraints.

This scheme generalizes and subsumes numerous classical methods, including block coordinate descent, expectation-maximization, and alternating minimization/projection. The key advantage lies in its ability to exploit problem structure for computational tractability.

2. Algorithmic Innovations and Extensions

2.1 Expanded and Joint Subspace Searches

Standard AO updates operate over coordinate (block-wise) subspaces. However, in nonconvex settings, this can trap iterates at saddle points or suboptimal minima due to the limited search directions. Innovations such as “expansion” phases (“escape” strategies) address this:

• Scaling (Perspective Variable) Approach: Rather than optimizing in the coordinate subspace for $z_b$, AO can be expanded by introducing a scaling variable $v_b$ for the complement ${z}_{-b}$, yielding a joint optimization over $(z_b, v_b)$:

$$\min_{z_b, v_b} f(v_b z_1, \ldots, z_b, \ldots, v_b z_B).$$

This technique enriches the update direction and aids in leaving undesirable stationary points (Murdoch et al., 2014).

• Restricted Joint Search: Updates are performed simultaneously in a restricted set of directions $w_1, ..., w_B$ (random, greedy, or data-dependent), as in:

$$\min_{\alpha_1, ..., \alpha_B} f(z_1 + \alpha_1 w_1, ..., z_B + \alpha_B w_B).$$

Joint, randomized, or data-adaptive subspace searches improve empirical convergence rates and final solution quality, particularly for problems such as matrix factorization and penalized regression involving MC+ penalties.

2.2 Distributed and Parallel Variants

Many large-scale problems possess block-separable structure, enabling distributed updates. In the context of trust region algorithms for linearly constrained nonlinear programs, the AO strategy is leveraged for blockwise “activity detection” via alternating projected gradient sweeps, followed by proximal regularized refinement steps. This two-phase decomposition allows effective distributed and parallel implementation, critical in applications such as optimal power flow (Hours et al., 2015).

Distributed AO methods often process local subproblems with local communication only, exploiting partial separability of the model. Proximal/trust region steps ensure robust and efficient convergence properties.

2.3 Inexact and Stochastic AO

AO schemes often encounter subproblems that are expensive or impossible to solve exactly. Inexact AO allows each block-update to be computed up to a specified tolerance or with controlled error sequences. When combined with stochastic approximations and variance reduction (e.g., SAGA, SARAH), AO can be scaled to very large, nonconvex, and nonsmooth problems without requiring full dataset computations per iteration (Driggs et al., 2020). Convergence to stationary points, and under additional conditions, to global optima can be established given mild decay or summability of errors (Pu et al., 2016, Driggs et al., 2020).

3. Convergence Properties and Sufficient Conditions

Theory for AO algorithms distinguishes between convex, nonconvex, and distributed/parallelized cases. Key facts include:

• Classical Convex Setting: For convex, coercive objectives and closed convex constraint sets, AO converges globally to a minimizer under broad conditions.
• Nonconvex Setting and Local Concavity: For loss functions $L(X,Y)$ minimized over nonconvex sets, such as rank-constrained matrix sets or sparse vectors, the convergence rate and guarantees depend critically on local concavity coefficients—a measure of the “degree of nonconvexity” around the solution (Ha et al., 2017). Sufficient conditions for linear convergence employ restricted strong convexity (RSC), restricted smoothness (RSM), and control on the local concavity coefficients to guarantee that the method converges linearly to a target up to statistical error, often at a rate governed by the better-conditioned block variable.
• Inexact AO: If subproblem solutions are computed inexactly (errors $\delta_x^t, \delta_y^t$), convergence can still be achieved provided these errors decay or are summable, potentially with the rate influenced by the marginal condition numbers of the respective subproblems.
• Multi-objective and Pareto Optimization: In the more general case of bi-objective (stochastic) alternating algorithms, convergence of the weighted sum of objectives is guaranteed at $O(1/T)$ under strong convexity, and $O(1/\sqrt{T})$ for convex objectives, with the flexibility to explore the Pareto front by adjusting step allocation per objective (Liu et al., 2022).

4. Algorithms for Structured Constraints and Penalties

AO methods are well suited for complex structured problems involving hard or soft constraints:

• Tensor and Matrix Factorization: For constrained CPD or PARAFAC2, AO alternates updates to factor matrices. Integration with primal-dual splitting allows decoupling of nonsmooth constraints and linear regularizers, eliminating the need for matrix inversion and accommodating hard (nonnegativity, unimodality) and soft (sparsity, TV, graph Laplacian) constraints (Ono et al., 2017, Roald et al., 2021).
• Discrete/Binary Decisions: For mixed Boolean nonlinear programs (e.g., load-shedding in microgrid optimization), AO decomposes into nonlinear programming for continuous variables and Boolean quadratic programming for discrete switches, alternating until convergence. Sequential convex approximation and penalty methods enforce nonconvex complementarity constraints, establishing local superlinear convergence properties (Du et al., 2022).

5. Quantum Alternating Optimization Algorithms

Quantum algorithms represent a major domain for AO-style methods. The Quantum Approximate Optimization Algorithm (QAOA) and its generalizations rely on an alternating application of “mixer” and “cost” unitaries:

• General Structure: QAOA alternates $p$ rounds of evolution under a cost Hamiltonian (diagonal encoding the objective) and a mixer Hamiltonian (driver for exploration of the search space). The quantum state is evolved as:

$$|\psi_p(\beta, \gamma)\rangle = \left(\prod_{l=1}^p e^{-i \beta_l B} e^{-i \gamma_l C} \right) |\psi_0\rangle$$

with $(\beta, \gamma)$ variationally tuned.

• Ansatz extensions: The Quantum Alternating Operator Ansatz (QAOA$^+$) and its variants employ mixing operators that enforce feasibility with respect to complex linear or combinatorial constraints—utilizing “merge” or “swap” operations rather than global bit flips (Hadfield et al., 2017, Goldstein-Gelb et al., 27 Sep 2024). Feasibility preservation, nonpositivity of the operator in the feasible subspace, and connectedness of the basis-state interaction graph are essential for convergence guarantees as $p \rightarrow \infty$.
• Design criteria and constraint handling: QAOA-style AO enables efficient encoding of hard combinatorial constraints (coloring, matching, scheduling) by designing mixers that restrict state evolution strictly within the feasible set, often yielding more efficient quantum circuits than penalization approaches (Hadfield et al., 2017, Chiang et al., 2022).

6. Application Domains and Practical Implications

AO is pervasive in applications where nonconvexity, large-scale variables, or structured constraints complicate direct optimization:

• Matrix/tensor factorization with structured regularization: AO with primal-dual or ADMM subroutines for each block enables flexible, scalable decomposition with rigorous guarantees for structured penalties (Ono et al., 2017, Roald et al., 2021).
• Quantum optimization and sampling: AO-form quantum algorithms solve unconstrained and linearly constrained combinatorial problems with provable convergence under realistic circuit depths (Hadfield et al., 2017, Goldstein-Gelb et al., 27 Sep 2024).
• Signal processing, distributed optimization, and control: AO with block-wise distribution and proximal regularization delivers scalable algorithms for nonconvex and linearly constrained power systems (Hours et al., 2015), distributed MPC, and consensus optimization (Pu et al., 2016).
• Multi-objective and fairness-constrained machine learning: Adjusting AO step allocation per objective yields scalable Pareto frontier exploration (Liu et al., 2022).
• Large-scale nonconvex learning: Stochastic AO with variance reduction (e.g., SPRING) provides computationally tractable methods for large imaging and sparse learning problems, with global convergence rates matching deterministic approaches (Driggs et al., 2020).

7. Future Directions and Generalizations

Recent research generalizes AO paradigms by:

• Expanding AO to highly nonconvex settings by dynamically constructing informed, data-driven escape directions.
• Developing hybrid classical-quantum AO frameworks for constrained optimization over discrete domains, with quantum mixing Hamiltonians ensuring feasibility (Goldstein-Gelb et al., 27 Sep 2024).
• Integrating AO with machine learning advances, including optimization-embedded neural network layers, which leverage AO-based alternating derivations for tractable and efficient layer differentiation (Sun et al., 2022).
• Exploring high-dimensional multi-block AO, stochastic and distributed settings, and theoretically characterizing Pareto–optimal strategies in multi-objective AO (Liu et al., 2022).

AO-based algorithm research continues to expand through hybridization with other algorithmic strategies, deeper exploitation of structural properties in application domains, and rigorous convergence analysis under broad conditions. The alternating update philosophy remains a versatile and theoretically robust foundation for scalable optimization in complex, high-dimensional environments.