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Last-iterate Convergence of ADMM on Multi-affine Quadratic Equality Constrained Problem

Yutong Chao, Michal Ciebielski, Jalal Etesami, Majid Khadiv

Abstract

In this paper, we study a class of non-convex optimization problems known as multi-affine quadratic equality constrained problems, which appear in various applications--from generating feasible force trajectories in robotic locomotion and manipulation to training neural networks. Although these problems are generally non-convex, they exhibit convexity or related properties when all variables except one are fixed. Under mild assumptions, we prove that the alternating direction method of multipliers (ADMM) converges when applied to this class of problems. Furthermore, when the "degree" of non-convexity in the constraints remains within certain bounds, we show that ADMM achieves a linear convergence rate. We validate our theoretical results through practical examples in robotic locomotion.

Last-iterate Convergence of ADMM on Multi-affine Quadratic Equality Constrained Problem

Abstract

In this paper, we study a class of non-convex optimization problems known as multi-affine quadratic equality constrained problems, which appear in various applications--from generating feasible force trajectories in robotic locomotion and manipulation to training neural networks. Although these problems are generally non-convex, they exhibit convexity or related properties when all variables except one are fixed. Under mild assumptions, we prove that the alternating direction method of multipliers (ADMM) converges when applied to this class of problems. Furthermore, when the "degree" of non-convexity in the constraints remains within certain bounds, we show that ADMM achieves a linear convergence rate. We validate our theoretical results through practical examples in robotic locomotion.
Paper Structure (18 sections, 17 theorems, 141 equations, 11 figures, 2 tables, 2 algorithms)

This paper contains 18 sections, 17 theorems, 141 equations, 11 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Setting
  3. Theoretical Results
  4. Application in Robotics
  5. Experiments
  6. Conclusion
  7. Technical Definitions and Lemmas
  8. Additional Experiment Details
  9. Derivations of the Locomotion Problem
  10. Proofs of theorems in Section \ref{['sec:theorem']}
  11. Approximated ADMM
  12. Error models.
  13. Proofs of Section \ref{['sec:robotics']}
  14. Additional Experiments Information
  15. Additional Experiments
  16. ...and 3 more sections

Key Result

Theorem 3.1

Suppose that ass_strongly_convex and ass_A_inside_Q hold. If $\phi$ is $L_\phi$-smooth, then ADMM in alg:admm with $\rho\geq\max\{\frac{4L_\phi^2}{\mu_\phi\lambda_{\min}^+(Q^TQ)},\frac{4L_\phi^2}{\mu_\phi\sqrt{\lambda_{\min}^+(QQ^T)}}\}$ converges with at least sublinear convergence rate to a statio where $x^\star=(x_1^\star, ..., x_n^\star)$ and for all $i$, and $z^\star$ is given by $z^\star\in

Figures (11)

  • Figure 1: Examples of locomotion and manipulation settings, (left) Solo grimminger2020open and (right) Trifinger wuthrich2020trifinger
  • Figure 2: Performance of the ADMM under the effect of nonlinearity.
  • Figure 3: Mean and standard deviation of dynamic violation values over optimization iterations. Results are shown for three different time discretizations. The x-axis shows the iteration number $k$.
  • Figure 4: Left: convex objective with multi-affine constraint. Center: convex objective with linear constraint. Right: nonconvex objective with linear constraint
  • Figure 5: Left: Schematic of a 2D locomotion problem. The robot has two contacts with friction $f_1$ and $f_2$. The location and angular momentum are $\mathbf{c}$ and $\mathbf{k}$. Center: Performance of the ADMM for different $\Delta t$. Right: Convergence rate of the ADMM for the 2D problem with random initialization.
  • ...and 6 more figures

Theorems & Definitions (32)

  • Definition 2.1
  • Example 2.2
  • Definition 2.4
  • Definition 2.5
  • Definition 2.7
  • Example 2.8
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Corollary 4.1
  • ...and 22 more