Application of interior point methods for non-linear programming

Abstract

In this thesis, we focus on the theoretical analysis and numerical investigation of specific interior point and conjugate gradient methods for solving nonlinear optimization problems. First, we propose logarithmic barrier approaches for constrained convex nonlinear optimization problems, where the barrier parameter is treated as a vector. This is followed by analytical studies in which the step-length is determined using the minorant function technique. The numerical findings reveal that the proposed methods exhibit both effectiveness and robustness. In addition, we develop new conjugate gradient methods for solving unconstrained optimization problems. These methods generate descent directions without requiring line search techniques. Moreover, they exhibit global convergence under mild assumptions. Numerical results indicate that the proposed methods are both effective and robust in addressing various unconstrained optimization and image restoration problems.