Locally Feasibly Projected Sequential Quadratic Programming for Nonlinear Programming on Arbitrary Smooth Constraint Manifolds

TL;DR

This paper introduces LFPSQP.jl, a Julia package implementing a feasible optimization algorithm for high-dimensional nonlinear problems on arbitrary smooth constraint manifolds, combining Riemannian optimization and SQP techniques.

Contribution

It presents a practical, efficient algorithm for feasible nonlinear optimization on arbitrary manifolds, integrating automatic differentiation and conjugate gradient methods.

Findings

Algorithm involves $O(nm^2)$-flop factorization per iteration.

Constructs efficient retractions with $O(nm)$-flop inner loop iterations.

Provides a Julia package leveraging automatic differentiation.

Abstract

High-dimensional nonlinear optimization problems subject to nonlinear constraints can appear in several contexts including constrained physical and dynamical systems, statistical estimation, and other numerical models. Feasible optimization routines can sometimes be valuable if the objective function is only defined on the feasible set or if numerical difficulties associated with merit functions or infeasible termination arise during the use of infeasible optimization routines. Drawing on the Riemannian optimization and sequential quadratic programming literature, a practical algorithm is constructed to conduct feasible optimization on arbitrary implicitly defined constraint manifolds. Specifically, with $n$ (potentially bound-constrained) variables and $m < n$ nonlinear constraints, each outer optimization loop iteration involves a single $O(nm^2)$-flop factorization, and computationally efficient retractions are constructed that involve $O(nm)$-flop inner loop iterations. A package, LFPSQP.jl, is created using the Julia language that takes advantage of automatic differentiation and projected conjugate gradient methods for use in inexact/truncated Newton steps.