[Paper Review] Distributed Semidefinite Programming with Application to Large-scale System Analysis

Sina Khoshfetrat Pakazad, Anders Hansson+2 more|arXiv (Cornell University)|Apr 29, 2015

Water resources management and optimization4 citations

TL;DR

This paper proposes a distributed primal-dual interior-point method for solving large-scale semidefinite programs (SDPs) with a tree-structured coupling pattern. By leveraging message-passing over a clique tree to compute exact search directions in finite steps, the algorithm achieves quadratic convergence with low per-iteration computational cost—factorizing small matrices and communicating only 12 times per agent per iteration—demonstrating superior efficiency over first-order methods in numerical tests on large-scale robustness analysis problems.

ABSTRACT

Distributed algorithms for solving coupled semidefinite programs (SDPs) commonly require many iterations to converge. They also put high computational demand on the computational agents. In this paper we show that in case the coupled problem has an inherent tree structure, it is possible to devise an efficient distributed algorithm for solving such problems. This algorithm can potentially enjoy the same efficiency as centralized solvers that exploit sparsity. The proposed algorithm relies on predictor-corrector primal-dual interior-point methods, where we use a message-passing algorithm to compute the search directions distributedly. Message-passing here is closely related to dynamic programming over trees. This allows us to compute the exact search directions in a finite number of steps. Furthermore this number can be computed a priori and only depends on the coupling structure of the problem. We use the proposed algorithm for analyzing robustness of large-scale uncertain systems distributedly. We test the performance of this algorithm using numerical examples.

Motivation & Objective

• To address the limitations of first-order distributed algorithms for large-scale SDPs, which suffer from slow convergence and high per-iteration computational load.
• To develop a distributed algorithm that leverages inherent tree structures in coupled SDPs to enable efficient, exact computation of search directions.
• To reduce computational burden per agent by distributing primal-dual interior-point iterations using dynamic programming-inspired message-passing.
• To enable robustness analysis of large-scale interconnected uncertain systems in a distributed, scalable, and efficient manner.
• To demonstrate that finite-step message-passing over trees allows exact search direction computation with a priori known complexity.

Proposed method

• The algorithm uses a predictor-corrector primal-dual interior-point method, distributing computations across agents via a tree-structured computational graph.
• Search directions are computed using a message-passing algorithm closely related to non-serial dynamic programming over the clique tree representation of the SDP's sparsity pattern.
• The message-passing process computes exact search directions in a finite number of steps, with the number of steps bounded by the tree's height and independent of problem size.
• Each agent performs only one matrix factorization per iteration and communicates with neighbors exactly 12 times per iteration, minimizing computational overhead.
• The method exploits chordal sparsity and clique tree decomposition to reformulate the coupled SDP into a distributed form amenable to parallel computation.
• The algorithm is applied to robustness analysis of large-scale interconnected uncertain systems, using numerical examples with scale-free network topologies.

Experimental results

Research questions

• RQ1 Can a distributed primal-dual interior-point method achieve faster convergence than first-order methods for large-scale SDPs with tree-structured coupling?
• RQ2 Can message-passing over a clique tree compute exact search directions for SDPs in finite steps, with complexity determined a priori?
• RQ3 What is the per-iteration computational cost and communication load per agent when distributing primal-dual iterations using tree structures?
• RQ4 How does the algorithm perform in practice on large-scale robustness analysis problems with sparse interconnections?
• RQ5 Can the proposed method scale efficiently to problems with high tree height and large numbers of cliques while maintaining quadratic convergence?

Key findings

• The algorithm converged in at most 14 iterations for a large-scale example with 579 cliques and a tree height of 35, demonstrating fast quadratic convergence.
• In the worst case, the algorithm required only 5,880 steps for the large-scale example, with each agent performing 14 matrix factorizations and 168 communications.
• For a smaller example with 198 subproblems and tree height 9, convergence occurred in 12 iterations, requiring 14,256 total steps.
• The number of required steps per agent was bounded and computable a priori, depending only on the tree height and not on the problem size.
• The convergence behavior closely matched that of standard centralized primal-dual methods, with a clear quadratic phase observed in numerical results.
• The method achieved exact search direction computation via message-passing, avoiding the approximation errors and slow convergence of first-order methods.