Inexact interior point methods for constrained convex quadratic optimization problems

This work proposes a new inexact interior point algorithm for convex quadratic programming problems with both equality and inequality constraints. Our method uses new preconditioned iterative methods to solve the linear systems that arise at every Newton step. These preconditioned conjugate gradient methods operate on an implicit Schur complement of the KKT system at each iteration. In contrast to standard approaches, the Schur complement we consider enables the reuse of the factorization of a fixed KKT subsystem across all interior point iterations. Further, the resulting reduced system admits preconditioners that directly alleviate the ill-conditioning associated with the strict complementarity condition in interior point methods. We propose two preconditioners that provably reduce the number of unique eigenvalues for the coefficient matrix (CG iteration count). One is efficient when the number of equality constraints is small, while the other is efficient when the number of remaining degrees of freedom is small. Numerical experiments with synthetic problems and problems from the Maros-Mészáros QP collection show that our preconditioned inexact interior point solvers are effective at improving conditioning and reducing cost relative to the best alternative preconditioned method for each problem. We first assume positive-definiteness of the Hessian then consider two additional cases distinguished by the invertibility of the KKT subsystem. We adapt the formulation of our inexact interior point method, as well as new preconditioners that account for the singularity of the Hessian, and the KKT subsystem respectively.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Computer Science

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Karim, Samah

Contributors dc:contributor

• Solomonik, Edgar
• Gropp, William
• Olson, Luke
• Chow, Edmond

• Primal-dual interior point methods
• KKT systems
• Krylov subspace methods
• Preconditioning

Statement dc:rights

• Copyright 2022 Samah Karim

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/116209

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F116209