# A gradient descent akin method for inequality constrained optimization

## TL;DR

This paper introduces a simplified gradient descent-like method for inequality constrained optimization, analyzing its convergence and demonstrating its effectiveness through numerical examples including shape optimization.

## Contribution

It presents a new computationally efficient first-order method derived from a previous search direction approach, with a rigorous mathematical analysis of its convergence properties.

## Key findings

- The method converges to local solutions under certain conditions.
- Numerical tests show the method's effectiveness on standard and shape optimization problems.
- The approach simplifies previous algorithms while maintaining convergence guarantees.

## Abstract

We propose a first-order method for solving inequality constrained optimization problems. The method is derived from our previous work [12], a modified search direction method (MSDM) that applies the singular-value decomposition of normalized gradients. In this work, we simplify its computational framework to a "gradient descent akin" method, i.e., the search direction is computed using a linear combination of the negative and normalized objective and constraint gradient. The main focus of this work is to provide a mathematical aspect to the method. We analyze the global behavior and convergence of the method using a dynamical systems approach. We then prove that the resulting trajectories find local solutions by asymptotically converging to the central path(s) for the logarithmic barrier interior-point method under the so-called relative convex condition. Numerical examples are reported, which include both common test examples and applications in shape optimization.