PGD-TO: A Scalable Alternative to MMA Using Projected Gradient Descent for Multi-Constraint Topology Optimization

Abstract

Projected Gradient Descent (PGD) methods offer a simple and scalable approach to topology optimization (TO), yet they often struggle with nonlinear and multi-constraint problems due to the complexity of active-set detection. This paper introduces PGD-TO, a framework that reformulates the projection step into a regularized convex quadratic problem, eliminating the need for active-set search and ensuring well-posedness even when constraints are infeasible. The framework employs a semismooth Newton solver for general multi-constraint cases and a binary search projection for single or independent constraints, achieving fast and reliable convergence. It further integrates spectral step-size adaptation and nonlinear conjugate-gradient directions for improved stability and efficiency. We evaluate PGD-TO on four benchmark families representing the breadth of TO problems: (i) minimum compliance with a linear volume constraint, (ii) minimum volume under a nonlinear compliance constraint, (iii) multi-material minimum compliance with four independent volume constraints, and (iv) minimum compliance with coupled volume and center-of-mass constraints. Across these single- and multi-constraint, linear and nonlinear cases, PGD-TO achieves convergence and final compliance comparable to the Method of Moving Asymptotes (MMA) and Optimality Criteria (OC), while reducing per-iteration computation time by 10-43x on general problems and 115-312x when constraints are independent. Overall, PGD-TO establishes a fast, robust, and scalable alternative to MMA, advancing topology optimization toward practical large-scale, multi-constraint, and nonlinear design problems. Public code available at: https://github.com/ahnobari/pyFANTOM