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An inertial ADMM for a class of nonconvex composite optimization with nonlinear coupling constraints

Le Thi Khanh Hien, Dimitri Papadimitriou

TL;DR

An inertial alternating direction method of multipliers for solving a class of non-convex multi-block optimization problems with nonlinear coupling constraints is proposed.

Abstract

In this paper, we propose an inertial alternating direction method of multipliers for solving a class of non-convex multi-block optimization problems with \emph{nonlinear coupling constraints}. Distinctive features of our proposed method, when compared with other alternating direction methods of multipliers for solving non-convex problems with nonlinear coupling constraints, include: (i) we apply the inertial technique to the update of primal variables and (ii) we apply a non-standard update rule for the multiplier by scaling the multiplier by a factor before moving along the ascent direction where a relaxation parameter is allowed. Subsequential convergence and global convergence are presented for the proposed algorithm.

An inertial ADMM for a class of nonconvex composite optimization with nonlinear coupling constraints

TL;DR

An inertial alternating direction method of multipliers for solving a class of non-convex multi-block optimization problems with nonlinear coupling constraints is proposed.

Abstract

In this paper, we propose an inertial alternating direction method of multipliers for solving a class of non-convex multi-block optimization problems with \emph{nonlinear coupling constraints}. Distinctive features of our proposed method, when compared with other alternating direction methods of multipliers for solving non-convex problems with nonlinear coupling constraints, include: (i) we apply the inertial technique to the update of primal variables and (ii) we apply a non-standard update rule for the multiplier by scaling the multiplier by a factor before moving along the ascent direction where a relaxation parameter is allowed. Subsequential convergence and global convergence are presented for the proposed algorithm.
Paper Structure (14 sections, 9 theorems, 87 equations, 1 figure, 2 tables, 1 algorithm)

This paper contains 14 sections, 9 theorems, 87 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notations
  3. Preliminaries
  4. Block surrogate and nearly sufficiently decrease property
  5. Augmented Lagrangian and stationary point
  6. Algorithm description and convergence analysis
  7. Algorithm description and the NSDPs
  8. Update rule for block $x_i$ and NSDP when updating $x_i$
  9. Update $y$
  10. Convergence analysis
  11. Sufficient decrease property
  12. Boundedness of subgradient
  13. Numerical results
  14. Conclusion

Key Result

proposition 1

Titan2020 Let $\Phi:\mathbb R^n\to\mathbb R$ be a lower semi-continuous function and $f_i:\mathbb R^n\to \mathbb R\cup \{+\infty\}$ be proper lower semi-continuous functions. Denote $x^{k,0} = x^k$, $x^{k,i}=(x^{k+1}_1, \ldots,x^{k+1}_{i},x^{k}_{i+1},\ldots,x^{k}_s)$ for $i \in [s]$, $x^{k+1} = x^{k If one of the following conditions holds: (note that $\rho_i(z)$ may depend on $z$), then we have

Figures (1)

  • Figure 1: Evolution of the log of mean of the objective values with respect to time.

Theorems & Definitions (20)

  • definition 1: Block surrogate function
  • proposition 1
  • proposition 2
  • definition 2
  • definition 3
  • proposition 3: NSDP when updating $x_i$
  • remark 1
  • remark 2
  • proposition 4: Sufficient decrease when updating $y$
  • proof
  • ...and 10 more