# Computing multiple solutions of topology optimization problems

arXiv: https://arxiv.org/abs/2004.11797 · HTML: https://mlsanity.com/paper/2004.11797/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Apr 2020.

## Built on by (most cited first, 7 of 7)

- [Neural topology optimization: the good, the bad, and the ugly](https://mlsanity.com/paper/2407.13954/) (2024)
- [Algebraic Language Models for Inverse Design of Metamaterials via Diffusion Transformers](https://mlsanity.com/paper/2507.15753/) (2025)
- [Learning Proximal Operators to Discover Multiple Optima](https://mlsanity.com/paper/2201.11945/) (2022)
- [Nonlinear Inverse Design of Mechanical Multi-Material Metamaterials Enabled by Video Denoising Diffusion and Structure Identifier](https://mlsanity.com/paper/2409.13908/) (2024)
- [A machine learning approach for fighting the curse of dimensionality in global optimization](https://mlsanity.com/paper/2110.14985/) (2021)
- [Simplifying deflation for non-convex optimization with applications in Bayesian inference and topology optimization](https://mlsanity.com/paper/2201.11926/) (2022)
- [Diverse Topology Optimization using Modulated Neural Fields](https://mlsanity.com/paper/2502.13174/) (2025)

## Abstract (arXiv)

Topology optimization problems often support multiple local minima due to a lack of convexity. Typically, gradient-based techniques combined with continuation in model parameters are used to promote convergence to more optimal solutions; however, these methods can fail even in the simplest cases. In this paper, we present an algorithm to perform a systematic exploratory search for the solutions of the optimization problem via second-order methods without a good initial guess. The algorithm combines the techniques of deflation, barrier methods and primal-dual active set solvers in a novel way. We demonstrate this approach on several numerical examples, observe mesh-independence in certain cases and show that multiple distinct local minima can be recovered.