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. Note that the 3 largest matrices (nlpkkt160, nlpkkt200, and nlpkkt240) require MATLAB 7.3 or later to read them.
Displaying all 5 collection matrices
|1901||nlpkkt80||Schenk||1,062,400||1,062,400||28,192,672||Optimization Problem||2008||MATLAB Rutherford Boeing Matrix Market|
|1902||nlpkkt120||Schenk||3,542,400||3,542,400||95,117,792||Optimization Problem||2008||MATLAB Rutherford Boeing Matrix Market|
|1903||nlpkkt160||Schenk||8,345,600||8,345,600||225,422,112||Optimization Problem||2008||MATLAB Rutherford Boeing Matrix Market|
|1904||nlpkkt200||Schenk||16,240,000||16,240,000||440,225,632||Optimization Problem||2008||MATLAB Rutherford Boeing Matrix Market|
|1905||nlpkkt240||Schenk||27,993,600||27,993,600||760,648,352||Optimization Problem||2008||MATLAB Rutherford Boeing Matrix Market|