A constrained optimization trajectory method using normalized gradients

We present a first-order method using normalized response gradients for constrained optimization problems. The method is consistently derived from our previous work, a modified search direction method developed for inequality constrained optimization problems that applies the singular-value decomposition (SVD). In the SVD-modified search direction method, a search direction is designed as a descent direction of the objective function. The resulting optimization trajectory converges to the central path of the interior-point method. The method has shown both efficiency and robustness in the industry-applicable structural optimization problems. However, there has been a lack of theory of the method, which this paper tries to address. Here, it will be shown that the formula for determining the search direction is surprisingly simple. The resulting optimization trajectory finds a local minimum for non-convex constrained optimization problems. We study the method in various examples, both analytically and numerically, and provide a convergence analysis of general 2D problems. An extension for general constrained optimization problems is given at the end of the paper, and an illustrative example is shown.