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Distributed Nonlinear Conic Optimisation with partially separable Structure

Richard Heusdens, Guoqiang Zhang

TL;DR

Distributed nonlinear conic optimization over graphs is tackled by extending the primal–dual method of multipliers (PDMM) to cone constraints via polar cones and a lifted-dual Peaceman-Rachford splitting, yielding a fully distributed update scheme. The paper unifies LP, QP, QCQP, SOCP, and SDP with partial separability under a single framework and provides convergence guarantees for both synchronous and stochastic updates. The core contribution is the generalized primal-dual method of multipliers (GPDMM) with explicit x and z updates, implemented in a distributed fashion through a lifted dual and local projections. Applications to cone-constrained consensus and decentralised max-cut SDP relaxation demonstrate scalability and practical viability for large-scale networked optimization problems.

Abstract

In this paper we consider the problem of distributed nonlinear optimisation of a separable convex cost function over a graph subject to cone constraints. We show how to generalise, using convex analysis, monotone operator theory and fixed-point theory, the primal-dual method of multipliers (PDMM), originally designed for equality constraint optimisation and recently extended to include linear inequality constraints, to accommodate for cone constraints. The resulting algorithm can be used to implement a variety of optimisation problems, including the important class of semidefinite programs with partially separable structure, in a fully distributed fashion. We derive update equations by applying the Peaceman-Rachford splitting algorithm to the monotonic inclusion related to the lifted dual problem. The cone constraints are implemented by a reflection method in the lifted dual domain where auxiliary variables are reflected with respect to the intersection of the polar cone and a subspace relating the dual and lifted dual domain. Convergence results for both synchronous and stochastic update schemes are provided and an application of the proposed algorithm is demonstrated to implement an approximate algorithm for maximum cut problems based on semidefinite programming in a fully distributed fashion.

Distributed Nonlinear Conic Optimisation with partially separable Structure

TL;DR

Distributed nonlinear conic optimization over graphs is tackled by extending the primal–dual method of multipliers (PDMM) to cone constraints via polar cones and a lifted-dual Peaceman-Rachford splitting, yielding a fully distributed update scheme. The paper unifies LP, QP, QCQP, SOCP, and SDP with partial separability under a single framework and provides convergence guarantees for both synchronous and stochastic updates. The core contribution is the generalized primal-dual method of multipliers (GPDMM) with explicit x and z updates, implemented in a distributed fashion through a lifted dual and local projections. Applications to cone-constrained consensus and decentralised max-cut SDP relaxation demonstrate scalability and practical viability for large-scale networked optimization problems.

Abstract

In this paper we consider the problem of distributed nonlinear optimisation of a separable convex cost function over a graph subject to cone constraints. We show how to generalise, using convex analysis, monotone operator theory and fixed-point theory, the primal-dual method of multipliers (PDMM), originally designed for equality constraint optimisation and recently extended to include linear inequality constraints, to accommodate for cone constraints. The resulting algorithm can be used to implement a variety of optimisation problems, including the important class of semidefinite programs with partially separable structure, in a fully distributed fashion. We derive update equations by applying the Peaceman-Rachford splitting algorithm to the monotonic inclusion related to the lifted dual problem. The cone constraints are implemented by a reflection method in the lifted dual domain where auxiliary variables are reflected with respect to the intersection of the polar cone and a subspace relating the dual and lifted dual domain. Convergence results for both synchronous and stochastic update schemes are provided and an application of the proposed algorithm is demonstrated to implement an approximate algorithm for maximum cut problems based on semidefinite programming in a fully distributed fashion.
Paper Structure (16 sections, 3 theorems, 65 equations, 4 figures, 1 algorithm)

This paper contains 16 sections, 3 theorems, 65 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main contribution
  3. Organisation of the paper
  4. Preliminaries
  5. Notations and functional properties
  6. Monotone operators and operator splitting
  7. Problem Setting
  8. Operater splitting of the lifted dual function
  9. Node constraints
  10. Local inequality constraints
  11. Convergence of the GPDMM algorithm
  12. Stochastic coordinate descent
  13. Applications and numerical experiments
  14. Cone constrained consensus problem
  15. Decentralised approximation algorithms for max-cut problems
  16. ...and 1 more sections

Key Result

Lemma 1

Let $y\in\bar{{{\cal G}}}$. Then

Figures (4)

  • Figure 1: Demonstration of a random geometric graph with 25 nodes.
  • Figure 2: Convergence results for GPDMM for the consensus problem \ref{['eq:fro']} over the graph depicted in Fig. \ref{['fig:graph_25nodes']} for (a) $(\forall i\in{{\cal V}}) \, K_i = {\mathbb{R}}_+^{5\times 10}$ and (b) $(\forall i\in{{\cal V}}) \, K_i = S^{10}_+$.
  • Figure 3: Convergence results for GPDMM for the max-cut problem. Fig. (a) shows convergence result of the primal variable, averaged over all entries of $X$ and all nodes, while (b) shows the distribution of 500 objective values for points sampled using randomised rounding.
  • Figure 4: Histogram of difference objective values.

Theorems & Definitions (12)

  • Definition 1: Monotone operator
  • Definition 2: Nonexpansiveness
  • Definition 3: Averaged nonexpansive operator
  • Lemma 1
  • proof
  • Example 1
  • Example 2
  • Proposition 1
  • proof
  • Remark 1
  • ...and 2 more