%-------------------------------------------------------------------------------
% UF Sparse Matrix Collection, Tim Davis
% http://www.cise.ufl.edu/research/sparse/matrices/Schenk/nlpkkt120
% name: Schenk/nlpkkt120
% [Symmetric indefinite KKT matrix, O. Schenk, Univ. of Basel]
% id: 1902
% date: 2008
% author: O. Schenk, A. Waechter, M. Weiser
% ed: T. Davis
% fields: name title A id date author ed kind Zeros notes
% kind: optimization problem
%-------------------------------------------------------------------------------
% notes:
% Symmetric indefinite KKT matrices, O. Schenk, Univ. of Basel,         
% Switzerland                                                           
% Nonlinear programming problems for a 3D PDE-constrained optimization  
% problem with boundary control as a function of the discretization     
% parameter N using 2nd-order finite difference approximations.         
%                                                                       
% O. Schenk, A. W\"achter, and M. Weiser, Inertia Revealing             
% Preconditioning For Large-Scale Nonconvex Constrained Optimization,   
% Technical Report, Unversity of Basel, 2008, submitted.                
%                                                                       
% Abstract: 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 1503 state variables and 1502 control variables, both   
% subject to bound constraints.                                         
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