Improving the Robustness of the Projected Gradient Descent Method for Nonlinear Constrained Optimization Problems in Topology Optimization

Abstract

The Projected Gradient Descent (PGD) algorithm is a widely used and efficient first-order method for solving constrained optimization problems due to its simplicity and scalability in large design spaces. Building on recent advancements in the PGD algorithm, where an inertial step component has been introduced to improve efficiency in solving constrained optimization problems, this study introduces two key enhancements to further improve the algorithm's performance and adaptability in large-scale design spaces. First, univariate constraints (such as design variable bounds constraints) are directly incorporated into the projection step via the Schur complement and an improved active set algorithm with bulk constraints manipulation, avoiding issues with min–max clipping. Second, the update step is decomposed relative to the constraint vector space, enabling a post-projection adjustment based on the state of the constraints and an approximation of the Lagrangian, significantly improving the algorithm's robustness for problems with nonlinear constraints. Applied to a topology optimization problem for heat sink design, the proposed PGD algorithm demonstrates performance comparable to or exceeding that of the Method of Moving Asymptotes (MMA), with minimal parameter tuning. These results position the enhanced PGD as a robust tool for complex optimization problems with large variable spaces, such as topology optimization problems.