In linear optimization, matrix structure can often be exploited algorithmically. However, beneficial presolving reductions sometimes destroy the special structure of a given problem. In this article, we discuss structure-aware implementations of presolving as part of a parallel interior-point method to solve linear programs with block-diagonal structure, including both linking variables and linking constraints. While presolving reductions are often mathematically simple, their implementation in a high-performance computing environment is a complex endeavor. We report results on impact, performance, and scalability of the resulting presolving routines on real-world energy system models with up to 700 million nonzero entries in the constraint matrix.
We investigate how to port the standard interior-point method to new exascale architectures for block-structured nonlinear programs with state equations. Computationally, we decompose the interior-point algorithm into two successive operations: the evaluation of the derivatives and the solution of the associated Karush-Kuhn-Tucker (KKT) linear system. Our method accelerates both operations using two levels of parallelism. First, we distribute the computations on multiple processes using coarse parallelism. Second, each process uses a SIMD/GPU accelerator locally to accelerate the operations us…
Interior point methods are among the most popular techniques for large scale nonlinear optimization, owing to their intrinsic ability of scaling to arbitrary large problem sizes. Their efficiency has attracted in recent years a lot of attention due to increasing demand for large scale optimization in industry and engineering. A parallel interior point method is discussed that exploits the intrinsic structure of large-scale nonlinear optimization problems so that the solution process can employ massively parallel high-performance computing infastructures. Since the overall performance of interi…
We propose an interior point method (IPM) for solving semidefinite programming problems (SDPs). The standard interior point algorithms used to solve SDPs work in the space of positive semidefinite matrices. Contrary to that the proposed algorithm works in the cone of matrices of constant \emph{factor width}. This adaptation makes the proposed method more suitable for parallelization than the standard IPM. We prove global convergence and provide a complexity analysis. Our work is inspired by a series of papers by Ahmadi, Dash, Majumdar and Hall, and builds upon a recent preprint by Roig-Solvas …
Due to critical environmental issues, the power systems have to accommodate a significant level of penetration of renewable generation which requires smart approaches to the power grid control. Associated optimal control problems are large-scale nonlinear optimization problems with up to hundreds of millions of variables and constraints. The interior point methods become computationally intractable, mainly due to the solution of large linear systems. This document addresses the computational bottlenecks of the interior point method during the solution of the security constrained optimal power …
In this paper, we consider the optimal coordination of automated vehicles at intersections under fixed crossing orders. We formulate the problem using direct optimal control and exploit the structure to construct a semi-distributed primal-dual interior-point algorithm to solve it by parallelizing most of the computations. Differently from standard distributed optimization algorithms, where the optimization problem is split, in our approach we split the linear algebra steps, such that the algorithm takes the same steps as a fully centralized one, while still performing computations in a distrib…
