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Distributed Computing for Huge-Scale Aggregative Convex Programming

Luoyi Tao

Abstract

Concerning huge-scale aggregative convex programming of a linear objective subject to the affine constraints of equality and inequality and the quadratic constraints of inequality, convex and aggregatively computable, an algorithm is developed for its distributed computing. The consensus with single common variable is used to partition the constraints into multi-consensus blocks, and the subblocks of each consensus block are employed to partition the primal variables into multiple sets of disjoint subvectors. The global consensus constraints of equality and the original constraints are converted into the extended constraints of equality via slack variables to help resolve feasibility and initialization of the algorithm. The augmented Lagrangian, the block-coordinate Gauss-Seidel method, the proximal point method with double proximal terms or single, and ADMM are used to update the primal and slack variables; descent models with built-in bounds are used to update the dual. Convergence of the algorithm to optimal solutions is argued and the rate of convergence, $O(1/k^{1/2})$ is estimated, under the feasibility supposed. Issues requiring further explorations are listed.

Distributed Computing for Huge-Scale Aggregative Convex Programming

Abstract

Concerning huge-scale aggregative convex programming of a linear objective subject to the affine constraints of equality and inequality and the quadratic constraints of inequality, convex and aggregatively computable, an algorithm is developed for its distributed computing. The consensus with single common variable is used to partition the constraints into multi-consensus blocks, and the subblocks of each consensus block are employed to partition the primal variables into multiple sets of disjoint subvectors. The global consensus constraints of equality and the original constraints are converted into the extended constraints of equality via slack variables to help resolve feasibility and initialization of the algorithm. The augmented Lagrangian, the block-coordinate Gauss-Seidel method, the proximal point method with double proximal terms or single, and ADMM are used to update the primal and slack variables; descent models with built-in bounds are used to update the dual. Convergence of the algorithm to optimal solutions is argued and the rate of convergence, is estimated, under the feasibility supposed. Issues requiring further explorations are listed.
Paper Structure (6 sections, 8 theorems, 70 equations)

This paper contains 6 sections, 8 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Formulation of optimization problem
  3. Update Rules for Primal Variables
  4. Dual Updates and Convergence Analysis
  5. Feasibility, Initialization, and Parameter Estimation
  6. Summary

Key Result

Lemma 1

The primal updates ACPGSProximalXilA, ACPGSProximalZl, ACPGSProximalsvZpiA through ACPGSProximalsvHniA yield, for all $k$, Here, $L^{k}$ is the augmented Lagrangian function sequence defined through the quantity $D^{k}$ is defined through which helps to develop the update rules for the duals, $\{\prescript{p\!X}{}{\!\mu}^{k+1}_{i}, \prescript{n\!X}{}{\!\mu}^{k+1}_{i}, \prescript{F\!}{}{\!\mu}^{

Theorems & Definitions (10)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Proof 1
  • Lemma 5
  • Proof 2
  • Lemma 6
  • Theorem 1
  • Theorem 2