A central path following method using the normalized gradients.

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.