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  <doc>
    <id>1031</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
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    <completedDate>2007-11-08</completedDate>
    <publishedDate>2007-11-08</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Inertia Revealing Preconditioning For Large-Scale Nonconvex Constrained Optimization</title>
    <abstract language="eng">Fast nonlinear programming methods following the all-at-once approach usually employ Newton's method for solving linearized Karush-Kuhn-Tucker (KKT) systems. In nonconvex problems, the Newton direction is only guaranteed to be a descent direction if the Hessian of the Lagrange function is positive definite on the nullspace of the active constraints, otherwise some modifications to Newton's method are necessary. This condition can be verified using the signs of the KKT's eigenvalues (inertia), which are usually available from direct solvers for the arising linear saddle point problems. Iterative solvers are mandatory for very large-scale problems, but in general do not provide the inertia. Here we present a preconditioner based on a multilevel incomplete $LBL^T$ factorization, from which an approximation of the inertia can be obtained. The suitability of the heuristics for application in optimization methods is verified on an interior point method applied to the CUTE and COPS test problems, on large-scale 3D PDE-constrained optimal control problems, as well as 3D PDE-constrained optimization in biomedical cancer hyperthermia treatment planning. The efficiency of the preconditioner is demonstrated on convex and nonconvex problems with $150^3$ state variables and $150^2$ control variables, both subject to bound constraints.</abstract>
    <identifier type="serial">07-32</identifier>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="opus3-id">1060</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-10314</identifier>
    <enrichment key="SourceTitle">Appeared in: SIAM J. Scientific Computing 31 (2008) 939-960</enrichment>
    <author>Olaf Schenk</author>
    <submitter>unknown unknown</submitter>
    <author>Andreas Wächter</author>
    <author>Martin Weiser</author>
    <series>
      <title>ZIB-Report</title>
      <number>07-32</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>nonconvex constrained optimization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>interior-point method</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>inertia</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>multilevel incomplete factorization</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="msc" number="49M15">Newton-type methods</collection>
    <collection role="msc" number="65F10">Iterative methods for linear systems [See also 65N22]</collection>
    <collection role="msc" number="65K10">Optimization and variational techniques [See also 49Mxx, 93B40]</collection>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="institutes" number="compmed">Computational Medicine</collection>
    <collection role="persons" number="weiser">Weiser, Martin</collection>
    <collection role="projects" number="MATHEON-A17">MATHEON-A17</collection>
    <collection role="projects" number="ZIB-Kaskade7">ZIB-Kaskade7</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/1031/ZR_07_32.pdf</file>
  </doc>
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