Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A first-order augmented Lagrangian method for constrained minimax optimization

Zhaosong Lu, Sanyou Mei

TL;DR

A first-order augmented Lagrangian method is proposed for solving a class of constrained minimax problems, whose subproblems turn out to be a much simpler structured minimax problem and are suitably solved by a first-order method developed in this paper.

Abstract

In this paper we study a class of constrained minimax problems. In particular, we propose a first-order augmented Lagrangian method for solving them, whose subproblems turn out to be a much simpler structured minimax problem and are suitably solved by a first-order method developed in this paper. Under some suitable assumptions, an \emph{operation complexity} of $O(\varepsilon^{-4}\log\varepsilon^{-1})$, measured by its fundamental operations, is established for the first-order augmented Lagrangian method for finding an $\varepsilon$-KKT solution of the constrained minimax problems.

A first-order augmented Lagrangian method for constrained minimax optimization

TL;DR

A first-order augmented Lagrangian method is proposed for solving a class of constrained minimax problems, whose subproblems turn out to be a much simpler structured minimax problem and are suitably solved by a first-order method developed in this paper.

Abstract

In this paper we study a class of constrained minimax problems. In particular, we propose a first-order augmented Lagrangian method for solving them, whose subproblems turn out to be a much simpler structured minimax problem and are suitably solved by a first-order method developed in this paper. Under some suitable assumptions, an \emph{operation complexity} of , measured by its fundamental operations, is established for the first-order augmented Lagrangian method for finding an -KKT solution of the constrained minimax problems.
Paper Structure (11 sections, 13 theorems, 140 equations, 3 algorithms)

This paper contains 11 sections, 13 theorems, 140 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notation and terminology
  3. A first-order method for nonconvex-concave minimax problem
  4. A modified optimal first-order method for strongly-convex-strongly-concave minimax problem
  5. A first-order method for problem \ref{['mmax-prob']}
  6. A first-order augmented Lagrangian method for problem \ref{['prob']}
  7. Complexity results for Algorithm \ref{['AL-alg']}
  8. Proof of the main result
  9. Proof of the main results in Subsection \ref{['strong-cvx-ccv']}
  10. Proof of the main results in Subsection \ref{['ppa']}
  11. Proof of the main results in Subsection \ref{['sec:complexity']}

Key Result

Theorem 1

Suppose that Assumptions a1 and ea hold. Let $\bar{H}^*$, $D_{\rm \bf x}$, $D_{\rm \bf y}$, $\bar{H}_{\rm low}$, and $\vartheta_0$ be defined in ea-prob, mmax-D, ea-bnd and ea-L, $\sigma_x$, $\sigma_y$ and $L_{\nabla \bar{h}}$ be given in Assumption ea, $\bar{\alpha}$, $\eta_y$, $\eta_z$, $\bar{\eps Then Algorithm mmax-alg1 outputs an $\bar{\epsilon}$-primal-dual stationary point of ea-prob in at

Theorems & Definitions (35)

  • Definition 1
  • Theorem 1: Complexity of Algorithm \ref{['mmax-alg1']}
  • Remark 1
  • Remark 2
  • Theorem 2: Complexity of Algorithm \ref{['mmax-alg2']}
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Definition 2
  • ...and 25 more