Exploiting Chordality in Optimization Algorithms for Model Predictive Control

Abstract

In this chapter we show that chordal structure can be used to devise efficient optimization methods for many common model predictive control problems. The chordal structure is used both for computing search directions efficiently as well as for distributing all the other computations in an interior-point method for solving the problem. The chordal structure can stem both from the sequential nature of the problem as well as from distributed formulations of the problem related to scenario trees or other formulations. The framework enables efficient parallel computations.

Parallel Exploitation for Tree-Structured Coupled Quadratic Programming in Julia

“…A simple computational graph which follows from the banded structure of the problem, is a graph with a chain of nodes. This, in fact, is the well-known backward dynamic programming formulation [2]. With this graph, however, we cannot benefit from parallelism since the computations in the nodes should be carried out sequentially.…”

“…The type of computational graphs which we are interested in, are the ones with a number of parallel branches, so that we can take advantage of parallelism. To this end, we use the approach proposed in [2], in which we can define computational graphs with arbitrary number of branches for the problem in (5). For details, see [2].…”

“…To this end, we use the approach proposed in [2], in which we can define computational graphs with arbitrary number of branches for the problem in (5). For details, see [2]. The idea in that paper is similar to what is presented in [8].…”

“…One application is MPC where an optimization problem needs to be solved repeatedly. In [2], the authors explore chordality for MPC problems and show how the banded structure stemming from an MPC problem can be exploited in order to find a computational graph for the distributed primal-dual algorithm which enables efficient parallel computations. In [8], a method is introduced for parallel computation of the Riccati recursion which in turn is used to solve constrained finitetime optimal control problems in parallel.…”

A dual Newton strategy for tree‐sparse quadratic programs and its implementation in the open‐source software treeQP

“…The search directions are calculated efficiently using a reverse structure‐exploiting Cholesky factorization with zero fill‐in outside the block structure and parallel processing of the tree branches. In the recent work of Hansson et al, it is argued that all the different structure‐exploiting schemes that have been proposed in the field of MPC are essentially based on the underlying chordal sparsity of their computational graphs (referred to as clique trees ). By applying the message passing algorithm of Pakazad et al on such graphs, one can obtain solution methods with the same complexity in a systematic way.…”