Motivated by the applications of the node-based shape optimization problem,
where various response evaluations are often considered in constrained
optimization, we propose a central path following method using the normalized
gradients. There exist numerous methods solving constrained optimization
problems. The methods of the class active-set strategy try to travel along the
active constraints to solve constrained optimization problems. Contrary to the
active-set methods, the algorithms of the class interior-point method reach an
optimal solution by traversing the interior of the feasible domain. This
characteristic is considered to be beneficial for shape optimization because
the usual zig-zagging behavior when traveling along the active constraints are
avoided. However, the interior-point methods require to solve a Newton problem
in each iteration, which is generally considered to be difficult for the shape
optimization problem. In the present work, we propose a path following method
based on the gradient flow calculated using the normalized gradients of the
objective and constraint functions. Applying the proposed method, we observe
that the centrality conditions for the interior-point method are approached
iteratively. The algorithm is able to approach a local minimum by traversing
the interior of the feasible domain by only using the gradient information of
the objective and constraint functions. We show a convergence analysis for a 2D
optimization problem with a linear constraint. The results are shown first with
analytical 2D constrained optimization problems, and then the results of shape
optimization problems with a large number of design variables are discussed. To
robustly deal with complex geometries, the Vertex Morphing method is used.