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Global weight optimization of frame structures with polynomial programming

Marek Tyburec, Michal Kočvara, Martin Kružík

TL;DR

This work addresses global weight optimization of frame structures with polynomial cross-section parametrization by leveraging the moment-sum-of-squares (MSOS) hierarchy to compute global minimizers. It extends previous MSOS results to provide feasible upper bounds derived from relaxed solutions and introduces a simple sufficient condition for global $\varepsilon$-optimality, with the guarantee of vanishing optimality gap when global minimizers form a convex set. A polynomial programming reformulation with quadratic bounds ensures MSOS convergence, and a univariate scalarization approach yields practical upper bounds. The approach is demonstrated on two small-scale examples, showing rapid convergence and the ability to certify optimality, with clear pathways for extending to dynamics and acceleration via sparsity and tameness techniques.

Abstract

Weight optimization of frame structures with continuous cross-section parametrization is a challenging non-convex problem that has traditionally been solved by local optimization techniques. Here, we exploit its inherent semi-algebraic structure and adopt the Lasserre hierarchy of relaxations to compute the global minimizers. While this hierarchy generates a natural sequence of lower bounds, we show, under mild assumptions, how to project the relaxed solutions onto the feasible set of the original problem and thus construct feasible upper bounds. Based on these bounds, we develop a simple sufficient condition of global $\varepsilon$-optimality. Finally, we prove that the optimality gap converges to zero in the limit if the set of global minimizers is convex. We demonstrate these results by means of two academic illustrations.

Global weight optimization of frame structures with polynomial programming

TL;DR

This work addresses global weight optimization of frame structures with polynomial cross-section parametrization by leveraging the moment-sum-of-squares (MSOS) hierarchy to compute global minimizers. It extends previous MSOS results to provide feasible upper bounds derived from relaxed solutions and introduces a simple sufficient condition for global -optimality, with the guarantee of vanishing optimality gap when global minimizers form a convex set. A polynomial programming reformulation with quadratic bounds ensures MSOS convergence, and a univariate scalarization approach yields practical upper bounds. The approach is demonstrated on two small-scale examples, showing rapid convergence and the ability to certify optimality, with clear pathways for extending to dynamics and acceleration via sparsity and tameness techniques.

Abstract

Weight optimization of frame structures with continuous cross-section parametrization is a challenging non-convex problem that has traditionally been solved by local optimization techniques. Here, we exploit its inherent semi-algebraic structure and adopt the Lasserre hierarchy of relaxations to compute the global minimizers. While this hierarchy generates a natural sequence of lower bounds, we show, under mild assumptions, how to project the relaxed solutions onto the feasible set of the original problem and thus construct feasible upper bounds. Based on these bounds, we develop a simple sufficient condition of global -optimality. Finally, we prove that the optimality gap converges to zero in the limit if the set of global minimizers is convex. We demonstrate these results by means of two academic illustrations.
Paper Structure (18 sections, 9 theorems, 33 equations, 2 figures, 2 tables)

This paper contains 18 sections, 9 theorems, 33 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Aims and novelty
  3. Moment-sum-of-squares hierarchy
  4. Methods
  5. Optimization problem formulation
  6. Properties of the compliance function
  7. Upper bounds to program \ref{['eq:sdp']} by scalarization
  8. Moment-sum-of-squares hierarchy
  9. Polynomial programming reformulation
  10. Recovering feasible upper bounds and sufficient condition of global $\varepsilon$-optimality
  11. Examples
  12. 24-element modular frame structure
  13. Part design
  14. Results and discussion
  15. Funding
  16. ...and 3 more sections

Key Result

theorem 1

Henrion2006 Let Assumption ass:archimedean be satisfied. Then, $f^{(r)}\nearrow f^{*}$ as $r \rightarrow \infty$.

Figures (2)

  • Figure 1: $24$-element frame structure: (a) boundary conditions, (b) cross-section parametrization, and (c) convergence of the proposed relaxation-based approach with visualized feasible upper-bound designs.
  • Figure 2: Frame structure reinforcement problem. (a) Discretization, boundary conditions and cross-section parametrization, and (b) convergence of the proposed relaxation-based approach with visualized feasible upper-bound designs.

Theorems & Definitions (21)

  • definition thmcounterdefinition
  • theorem 1
  • proposition thmcounterproposition
  • proof
  • proposition thmcounterproposition
  • proof
  • remark thmcounterremark
  • proposition thmcounterproposition
  • proof
  • proposition thmcounterproposition
  • ...and 11 more