Term-sparse polynomial optimization for the design offrame structures

This work investigates an efficient solution to two fundamental problems in topology optimization of frame structures. The first one involves minimizing structural compliance under linear-elastic equilibrium and weight constraint, while the second one minimizes the weight under compliance constraints. These problems are non-convex and generally challenging to solve globally, with the non-convexity concentrated in a polynomial matrix inequality. In Tyburec et al. (2021, 2023), the authors tackled the problems using the moment-sum-of-squares hierarchy (mSOS), but were only able to solve smaller instances globally. Here, we aim to improve the scalability of solution to these problems by using the mSOS hierarchy supplemented with the Term Sparsity Pattern technique (TSP), which was introduced by Magron and Wang (2023). Due to the unique polynomial structure of our problems in which the objective and constraint functions are separable polynomials, we further improve scalability by adopting a reduced monomial basis containing non-mixed terms only. From extensive numerical testing, we conclude that these techniques allow for a global solution to two times larger instances when compared to (Tyburec et al. 2021, 2023), and accelerate the solution of the problems from Tyburec et al. (2021, 2023) significantly.