EURO 2024 Copenhagen
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1627. Weight minimization of frame structures under free-vibration eigenvalue constraints with polynomial optimization

Invited abstract in session WD-38: Advances in polynomial optimization and its applications, stream Conic Optimization: Theory, Algorithms, and Applications.

Wednesday, 14:30-16:00
Room: 34 (building: 306)

Authors (first author is the speaker)

1. Marek Tyburec
Department of Mechanics, Czech Technical University in Prague, Faculty of Civil Engineering

Abstract

Topology optimization of frame structures that minimizes the weight while respecting free-vibration eigenvalue constraints presents a challenging nonconvex polynomial optimization problem. We address this challenge using a nonlinear semidefinite programming formulation, casting it as a bilevel program. This reformulation offers a unique structure: a quasiconvex, univariate lower-level problem that searches for scaling of fixed cross-section area ratios and an upper level that optimizes these ratios. We establish conditions for the lower-level problem solvability and demonstrate how to construct feasible solutions for the original polynomial semidefinite program. Given a feasible point, we prove that the convergence conditions of the Lasserre hierarchy are satisfied. By solving this hierarchy of convex relaxations, we obtain lower bounds. We further show that the relaxed solutions satisfy the conditions for the feasibility of the lower-level problem and thus enable us to construct feasible upper bounds in each relaxation. With this, we introduce a simple sufficient condition for global ε-optimality and prove that the optimality gap ε approaches zero in the limit when the set of global minimizers is convex. Numerical examples illustrate the convergence of the hierarchy in a finite number of steps.

Keywords

Status: accepted

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