Inexact interior point methods for constrained convex quadratic optimization problems

Karim, Samah

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

Description

Title

Inexact interior point methods for constrained convex quadratic optimization problems

Author(s)

Karim, Samah

Issue Date

2022-07-12

Director of Research (if dissertation) or Advisor (if thesis)

Solomonik, Edgar

Doctoral Committee Chair(s)

Solomonik, Edgar

Committee Member(s)

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

Department of Study

Computer Science

Discipline

Computer Science

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

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

Abstract

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.

Graduation Semester

2022-08

Type of Resource

Thesis

Copyright and License Information

Copyright 2022 Samah Karim

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