---
title: Tensor-Based Proximal Alternating Minimization
url: https://www.emergentmind.com/topics/tensor-based-proximal-alternating-minimization-algorithm
type: topic
---

# Tensor-Based Proximal Alternating Minimization

A tensor-based proximal alternating minimization (PAM) algorithm is a numerical optimization approach developed for solving inhomogeneous quartic polynomial optimization problems on the sphere, with a structure inspired by applications such as discretized Bose–Einstein condensate (BEC) ground state computations. The method constructs an equivalence between a fourth-degree inhomogeneous polynomial minimization and a four-block multilinear optimization problem (MOP), exploiting tensor representations and block coordinate descent with proximal regularization for efficient, closed-form iterative updates. The convergence of the algorithm is established under mild assumptions, and empirical studies demonstrate notable gains compared to alternative methods such as inexact ADMM [2512.24872].

## 1. Problem Formulation and Tensor-Multilinear Equivalence

The method addresses minimization of an inhomogeneous quartic polynomial of the form:
\[
\min_{x\in\R^n}\; f(x)\;=\;\frac{\theta}{2}\sum_{i=1}^n x_i^4\;+\;x^T B x \quad\text{s.t.}\;\|x\|=1,
\]
where $\theta > 0$ and $B\in\R^{n\times n}$ is symmetric. Any degree-4 polynomial $g(x)$ in $n$ variables can be expressed via a symmetric order-4 tensor $\mathcal{T}$ as $g(x)=\langle \mathcal{T},x\circ x\circ x\circ x\rangle$. In the inhomogeneous case, the variable is lifted to $\tilde x=(1, x^T)^T \in \R^{n+1}$, and the homogenized tensor $\mathcal{T}_f\in\S^{4,n+1}$ encodes the full objective.

Defining a corresponding four-way multilinear function,
\[
F(x,y,z,w) = \langle \mathcal{T}_f, \tilde x\circ\tilde y\circ\tilde z\circ\tilde w\rangle,
\]
with each $\tilde x, \tilde y, \tilde z, \tilde w \in \R^{n+1}$ (first entry fixed at 1, remainder from the respective variables), a key result is that, assuming a mild concavity condition on $u\mapsto\mathcal{T}_f u^4$, minimization of the original quartic is equivalent to minimizing $F$ over four blocks constrained to the unit sphere:
\[
\min_{\|x\|=1} f(x) = \min_{\|x\|=\|y\|=\|z\|=\|w\|=1} F(x, y, z, w).
\]
To guarantee global concavity, a shift $-\alpha\|\tilde x\|^4$ is introduced, with $\alpha \geq \|\mathcal{T}_f\|$. The modified problem is
\[
f_\alpha(x) = \langle \mathcal{T}_f, \tilde x^4 \rangle - \alpha \|\tilde x\|^4,
\]
and its multilinear equivalent is
\[
F_\alpha(x, y, z, w) = \langle \mathcal{T}_f, \tilde x\circ\tilde y\circ\tilde z\circ\tilde w \rangle - \alpha \langle \tilde x, \tilde y \rangle \langle \tilde z, \tilde w \rangle,
\]
with the minimizers of both formulations coinciding under these conditions [2512.24872].

## 2. Multi-Block Structure and Blockwise Minimization

The MOP
\[
\min_{\|x\|=\|y\|=\|z\|=\|w\|=1} F_\alpha(x, y, z, w)
\]
exhibits a four-block structure, each block constrained to the unit sphere. By freezing three blocks, the subproblem for the remaining block reduces to a quartic–quadratic function composed with a proximal term. Multilinearity ensures that block coordinate descent (BCD) delivers closed-form updates at each step.

This natural cyclic update scheme $(x \to y \to z \to w)$ is the basis for the overall proximal alternating minimization algorithm.

## 3. Proximal Alternating Minimization Updates

Defining $\mathcal{G} = \{ \tilde x=(1, x^T)^T : \|x\|=1 \}$, the PAM algorithm performs, for $k=0,1,2,\dots$,
\[
\begin{cases}
\tilde x^{(k+1)} = \displaystyle\arg\min_{\tilde x \in \mathcal{G}} \left\{ F_\alpha(x, y^{(k)}, z^{(k)}, w^{(k)}) + \frac{\gamma_1}{2}\|\tilde x - \tilde x^{(k)}\|^2 \right\}, \\
\tilde y^{(k+1)} = \displaystyle\arg\min_{\tilde y \in \mathcal{G}} \left\{ F_\alpha(x^{(k+1)}, y, z^{(k)}, w^{(k)}) + \frac{\gamma_2}{2}\|\tilde y - \tilde y^{(k)}\|^2 \right\}, \\
\tilde z^{(k+1)} = \displaystyle\arg\min_{\tilde z \in \mathcal{G}} \left\{ F_\alpha(x^{(k+1)}, y^{(k+1)}, z, w^{(k)}) + \frac{\gamma_3}{2}\|\tilde z - \tilde z^{(k)}\|^2 \right\}, \\
\tilde w^{(k+1)} = \displaystyle\arg\min_{\tilde w \in \mathcal{G}} \left\{ F_\alpha(x^{(k+1)}, y^{(k+1)}, z^{(k+1)}, w) + \frac{\gamma_4}{2}\|\tilde w - \tilde w^{(k)}\|^2 \right\}.
\end{cases}
\]
Each block update is a strongly convex quadratic minimization on the sphere, with a closed-form solution:
\[
x^{(k+1)} = \pm \frac{g_x^{(k)}-\gamma_1 x^{(k)}}{\|g_x^{(k)}-\gamma_1 x^{(k)}\|},
\]
where
\[
g_x^{(k)} = (I_{[2:n+1]}) \langle \mathcal{T}_f, \tilde y^{(k)} \circ \tilde z^{(k)} \circ \tilde w^{(k)} \rangle - \alpha \langle \tilde z^{(k)},\tilde w^{(k)} \rangle y^{(k)}.
\]
Here, $I_{[2:n+1]}$ selects the relevant indices in the lifted variable. The regularization parameters $\gamma_i > 0$ ensure the strong convexity of each subproblem.

## 4. Convergence Properties

Key convergence assumptions include (i) $\alpha \geq \|\mathcal{T}_f\|$ for concavity, (ii) positive proximal coefficients $\gamma_i$, and (iii) compactness of the feasible set $\mathcal{G}^4$. Theoretical results (Theorem 4.1 and 4.2) establish:

1. **Descent**: After each full cycle, the objective $F_\alpha$ decreases at least by a multiple of the squared-step norm, guaranteeing monotonic convergence.
2. **Vanishing Steps**: The sum of squared step sizes is finite; thus, $\|\tilde t^{(k+1)} - \tilde t^{(k)}\| \to 0$ as $k\to\infty$ (with $\tilde t^{(k)}$ collecting all lifted variables).
3. **Cluster Points**: Compactness ensures existence of accumulation points.
4. **Stationarity**: Any cluster point satisfies the KKT conditions for the MOP, i.e., each block’s variational inequality for $F_\alpha$.

## 5. Computational Complexity Per Iteration

Each block update's primary expense is the tensor contraction $\langle\mathcal{T}_f, \tilde u\circ\tilde v\circ\tilde w\rangle$. For a general dense order-4 tensor, this cost scales as $O(n^4)$. Nevertheless, when $\mathcal{T}_f$ arises from BEC applications, e.g., via discretized Gross–Pitaevskii energies, the tensor exhibits significant sparsity and symmetry; the dominant cost then reduces to $O(n^2)$ or even $O(n)$ per contraction for 1D and 2D finite-difference grids.

Each block update further requires $O(n)$ operations for normalization and vector addition, so, in practice, total cost per full iteration is $O(n^2)$ in 1D and can approach $O(n)$ in 2D with optimal exploitation of tensor structure.

## 6. Hyperparameter Selection and Acceleration

Several algorithmic choices affect performance:

- **Shift parameter $\alpha$**: Chosen to satisfy $\alpha \geq \|\mathcal{T}_f\| \approx \max|\mathcal{T}_f u^4|^{1/4}$. Slightly larger $\alpha$ improves concavity and numerical stability but excessive scaling suppresses the multilinear term, slowing convergence.
- **Proximal coefficients $\gamma_i$**: Any positive value yields convergence. In practice, a moderate constant (e.g., $0.5$) balances progress and stability.
- **Initialization**: Random unit-norm initialization suffices; marginally improved warm starts may be obtained via a few steps of the power method or MBI.
- **Acceleration strategies**: Proximal coefficients may be increased adaptively when little progress is detected, or SQUAREM-type extrapolation may be applied to the iterates for superlinear convergence in early phases.

## 7. Numerical Experiments and Performance Comparison

Empirical validation employed synthetic BEC data, with both 1D and 2D Gross–Pitaevskii energy discretizations. The algorithm was compared to an inexact ADMM (with Newton-type inner iterations), across a range of discretization sizes:

- **Iteration and runtime efficiency**: PAM required $4$–$20$ outer iterations and $0.001$–$0.05$ seconds, while ADMM took $50$–$300$ iterations and $0.02$–$0.2$ seconds, with both methods attaining identical objective values to machine accuracy.
- **Convergence profile**: PAM's objective exhibited rapid initial decrease and smooth convergence, contrasting with the slower final-phase descent in ADMM, due to inexact Newton steps.
- **Ground-state recovery**: Final minimizers reproduced nonnegative, symmetric, and exponentially decaying ground-state wavefunctions, in agreement with analytical expectations for BEC.
- **Sensitivity to $\alpha$**: Modest variation of $\alpha$ from $\|\mathcal{T}_f\|$ to $3\|\mathcal{T}_f\|$ had little impact on final objectives, with runtime varying by at most a factor of two. Setting $\alpha<\|\mathcal{T}_f\|$ generated instability; large $\alpha$ slowed progress.

In sum, the tensor-based PAM method exploits the equivalence between inhomogeneous quartic and multilinear formulations to provide efficient blockwise updates with simple convergence guarantees, demonstrating superior per-iteration efficacy compared to ADMM in the context of synthetic BEC tests [2512.24872].

Source: https://www.emergentmind.com/topics/tensor-based-proximal-alternating-minimization-algorithm