Interior Point (IP) methods are very effective techniques for solving several classes of optimization problems. IP methods are iterative methods requiring at each iteration to solve a linear system in order to compute a search direction. For large scale problems it is expensive to compute such directions; moreover, it may be unnecessary to compute them with a high accuracy if we are far from a solution. As a result it can be convenient to use iterative methods for solving the linear system with an accuracy which increases as far as we get closer to the solution. By using iterative methods we obtain Inexact (or Truncated) IP methods.
We shall present an infeasible inexact path-following IP method for solving the Nonlinear Complementarity Problem. The method compute inexactly a search direction by performing an Inexact Newton step at each iteration. Then, a suitable stepsize is searched in order to satisfy some classical centering conditions and an Armijo rule for a given merit function. Under proper assumptions fast global convergence is ensured.
We shall also propose possible application of Inexact Interior Point methods so Semidefinite Programming, pointing out some open questions concerning linear algebra issues.
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