Large-scale Optimization with Linear Equality Constraints using Reduced Compact Representation

TL;DR

This paper introduces a novel reduced compact representation for large-scale optimization problems with linear equality constraints, improving computational efficiency by leveraging nullspace projections and sparse matrix techniques.

Contribution

It develops a new reduced compact representation of the inverse Hessian for constrained optimization, enabling faster search direction computations in large-scale problems.

Findings

Significant reduction in computation times for trust-region algorithms

Improved efficiency over previous implementations and IPOPT

Effective use of sparse QR and LSQR for projections

Abstract

For optimization problems with linear equality constraints, we prove that the (1,1) block of the inverse KKT matrix remains unchanged when projected onto the nullspace of the constraint matrix. We develop reduced compact representations of the limited-memory inverse BFGS Hessian to compute search directions efficiently when the constraint Jacobian is sparse. Orthogonal projections are implemented by a sparse QR factorization or a preconditioned LSQR iteration. In numerical experiments two proposed trust-region algorithms improve in computation times, often significantly, compared to previous implementations of related algorithms and compared to IPOPT.