Diploma thesis as pdf file - Johannes Kepler University, Linz

Chapter 2
Interior point method for the
numerical solution of the obstacle
problem
2.1 The central path
The idea of the interior point methods is to replace the constrained minimization
problem by a sequence of unconstrained minimization problems, for solving of
which we can use Newton’s method. An objective functional of the unconstrained
problem we generate by adding barrier functional to the objective functional of the
original constrained problem. Barrier function serves as barrier against leaving of
the elements the feasible region K. Each of the problems in the sequence corresponds
to the objective functional J κ (v) depending on the nonnegative penalty
parameter κ. For our problem we construct J κ (v) in the following way:
∫
J κ (v) = J(v) − κ ln(v − ψ)dω.
Ω
We extend the definition of the ln to the whole real domain axis by setting lnz =
−∞,
z ≤ 0. Then barrier function approaches infinity as the elements from the
interior approach the boundary. Thus we obtained the family of the following
unconstrained minimization problem:
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