A study of one-parameter regularization methods for mathematical programs with vanishing constraints

ABSTRACT

Mathematical programs with vanishing constraints (MPVCs) are a class of nonlinear optimization problems with applications to various engineering problems such as truss topology design and robot motion planning. MPVCs are difficult problems from both a theoretical and numerical perspective: the combinatorial nature of the vanishing constraints often prevents standard constraint qualifications and optimality conditions from being attained; moreover, the feasible set is inherently nonconvex, and often has no interior around points of interest. In this paper, we therefore study and compare four regularization methods for the numerical solution of MPVCs. Each method depends on a single regularization parameter, which is used to embed the original MPVC into a sequence of standard nonlinear programs. Convergence results for these methods based on both exact and approximate stationary of the subproblems are established under weak assumptions. The improved regularity of the subproblems is studied by providing sufficient conditions for the existence of KKT multipliers. Numerical experiments, based on applications in truss topology design and an optimal control problem from aerothermodynamics, complement the theoretical analysis and comparison of the regularization methods. The computational results highlight the benefit of using regularization over applying a standard solver directly, and they allow us to identify two promising regularization schemes.

Keywords:

• Mathematical program with vanishing constraints
• constraint qualification
• optimality conditions
• M-stationarity
• T-stationarity
• regularization method
• truss topology optimization
• aerothermodynamics

2010 Mathematics Subject Classifications:

• 65K05
• 90C30
• 90C31
• 49J15
• 49M20

Acknowledgments

The authors would like to thank two anonymous referees whose comments led to an improvement of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Here, for all convergence results, the point $x^{\ast }$ in question is always feasible for the underlying MPVC, so that the imposed constraint qualifications at $x^{\ast }$ are well-defined.

Additional information

Funding

The work of T. Hoheisel was supported by the Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada Discovery Grant RGPIN-2017-04035. The work of B. Pablos was supported by the Bavarian Research Alliance and Munich Aerospace. The work of A. Pooladian was supported by NSERC CGS-M and Lorne Trottier Accelerator Fellowship. The work of A. Schwartz was supported by the Excellence Initiative of the German Federal and State Governments and the Graduate School of Computational Engineering at Technische Universität Darmstadt. The work of L. Steverango was supported by an ISM Undergraduate Summer Research Scholarship.

Notes on contributors

Tim Hoheisel

Tim Hoheisel received a doctorate degree in mathematics from Julius-Maximilians University, Würzburg in 2009. He held a postdoctoral position in Würzburg until 2016. During this time he was a visiting professor at Heinrich-Heine University, Düsseldorf in 2011/12, and a visiting scholar at University of Washington, Seattle in 2012 and 2014. He joined the Department of Mathematics and Statistics at McGill University, Montreal in 2016 as an assistant professor. His research area lies at the interface of continuous optimization and nonsmooth analysis.

Blanca Pablos

Blanca Pablos, at the time of publication, is a PhD student at the Engineering Mathematics Department of the University of the Armed Forces in Munich. She is a member of the Munich Aerospace research group “Re-entry optimization to minimize heating”. Her research focuses on optimal control techniques applied to the problem of atmospheric re-entry.

Aram Pooladian

Aram Pooladian, at the time of publication, is a Master's student at McGill University supervised by Professor Tim Hoheisel and Adam Oberman committed to starting his PhD at Center for Data Science at New York University. His research interests lie at the intersection of optimization theory, computational and statistical optimal transport, high dimensional statistics, and deep learning.

Alexandra Schwartz

Alexandra Schwartz received a doctorate degree in mathematics from Julius-Maximilians University, Würzburg in 2011. She held a postdoc position in Würzburg until 2014 when she was appointed Junior Professor at the Department of Mathematics at the Technical University of Darmstadt. In 2019 she was a visiting W2-professor at the Technical University of Dresden. Her research interests cover game theory, bilevel optimization and nonlinear programming with combinatorial constraints.

Luke Steverango

Luke Steverango's contribution to this publication was made when he was finishing up his Undergraduate Honours degree in Mathematics at McGill University. He has since gone on to study at Queen's University for his Master's degree under the supervision of Professor Peter Taylor. He will continue on at Queen's University under the direction of Professor Catherine Pfaff for his PhD. His interests lie in applications of topology, topological data analysis, and geometric modelling in sports analytics.