Dynamic optimization with convergence guarantees
File(s)
Working paper
Author(s)
Neuenhofen, Martin P
Kerrigan, Eric C
Type
Working Paper
Abstract
We present a novel direct transcription method to solve optimization problems
subject to nonlinear differential and inequality constraints. In order to
provide numerical convergence guarantees, it is sufficient for the functions
that define the problem to satisfy boundedness and Lipschitz conditions. Our
assumptions are the most general to date; we do not require uniqueness,
differentiability or constraint qualifications to hold and we avoid the use of
Lagrange multipliers. Our approach differs fundamentally from state-of-the-art
methods based on collocation. We follow a least-squares approach to finding
approximate solutions to the differential equations. The objective is augmented
with the integral of a quadratic penalty on the differential equation residual
and a logarithmic barrier for the inequality constraints, as well as a
quadratic penalty on the point constraint residual. The resulting unconstrained
infinite-dimensional optimization problem is discretized using finite elements,
while integrals are replaced by quadrature approximations if they cannot be
evaluated analytically. Order of convergence results are derived, even if
components of solutions are discontinuous.
subject to nonlinear differential and inequality constraints. In order to
provide numerical convergence guarantees, it is sufficient for the functions
that define the problem to satisfy boundedness and Lipschitz conditions. Our
assumptions are the most general to date; we do not require uniqueness,
differentiability or constraint qualifications to hold and we avoid the use of
Lagrange multipliers. Our approach differs fundamentally from state-of-the-art
methods based on collocation. We follow a least-squares approach to finding
approximate solutions to the differential equations. The objective is augmented
with the integral of a quadratic penalty on the differential equation residual
and a logarithmic barrier for the inequality constraints, as well as a
quadratic penalty on the point constraint residual. The resulting unconstrained
infinite-dimensional optimization problem is discretized using finite elements,
while integrals are replaced by quadrature approximations if they cannot be
evaluated analytically. Order of convergence results are derived, even if
components of solutions are discontinuous.
Date Issued
2018-10-09
Citation
2018
Publisher
arXiv
Copyright Statement
© 2018 The Author(s)
Identifier
http://arxiv.org/abs/1810.04059v1
Subjects
math.OC
math.OC
cs.SY
Publication Status
Published