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Term-sparse polynomial optimization for the design of frame structures

Marouan Handa, Marek Tyburec, Michal Kočvara

TL;DR

This work tackles global optimization for frame-structure topology design, where the governing PMI induces non-convexity. It leverages Lasserre's mSOS hierarchy, augmented with the Term Sparsity Pattern (TSP) and a reduced non-mixed monomial basis (NMT) that matches the separable polynomial structure of frame problems. The authors adapt TSP to polynomial matrices, provide ε-optimality certificates, and demonstrate substantial scalability gains, solving problems with up to 39 elements and achieving global optima orders of magnitude faster than dense relaxations in many cases. While the NMT approach shows strong empirical performance, it lacks a general convergence proof; the work outlines future directions to further exploit stiffness sparsity and chordal decompositions for even larger and more complex designs. Overall, the paper contributes practical, scalable tools for certifying global optima in frame-structure optimization with significant implications for reliable, high-performance structural design.

Abstract

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. We solve these problems using the moment-sum-of-squares (mSOS) hierarchy and improve the scalability by enhancing (mSOS) with the Term Sparsity Pattern (TSP) technique. Additionally, we exploit the unique polynomial structure of our problems by adopting a reduced monomial basis containing only non-mixed terms. These modifications significantly enhance computational efficiency. Extensive numerical experiments demonstrate that our approach achieves global solutions for instances twice as large as those previously solved while substantially accelerating the solution process.

Term-sparse polynomial optimization for the design of frame structures

TL;DR

This work tackles global optimization for frame-structure topology design, where the governing PMI induces non-convexity. It leverages Lasserre's mSOS hierarchy, augmented with the Term Sparsity Pattern (TSP) and a reduced non-mixed monomial basis (NMT) that matches the separable polynomial structure of frame problems. The authors adapt TSP to polynomial matrices, provide ε-optimality certificates, and demonstrate substantial scalability gains, solving problems with up to 39 elements and achieving global optima orders of magnitude faster than dense relaxations in many cases. While the NMT approach shows strong empirical performance, it lacks a general convergence proof; the work outlines future directions to further exploit stiffness sparsity and chordal decompositions for even larger and more complex designs. Overall, the paper contributes practical, scalable tools for certifying global optima in frame-structure optimization with significant implications for reliable, high-performance structural design.

Abstract

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. We solve these problems using the moment-sum-of-squares (mSOS) hierarchy and improve the scalability by enhancing (mSOS) with the Term Sparsity Pattern (TSP) technique. Additionally, we exploit the unique polynomial structure of our problems by adopting a reduced monomial basis containing only non-mixed terms. These modifications significantly enhance computational efficiency. Extensive numerical experiments demonstrate that our approach achieves global solutions for instances twice as large as those previously solved while substantially accelerating the solution process.

Paper Structure

This paper contains 27 sections, 5 theorems, 46 equations, 13 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Contribution and the organization of the paper
  3. Background: topology optimization of frame structures
  4. Formulation
  5. Illustration problem
  6. Lasserre's moment sum-of-square hierarchy
  7. Definitions and notations
  8. The moment sum-of-squares hierarchy
  9. Certificate of global $\varepsilon$-optimality in frame structures optimization
  10. Compliance optimization
  11. Weight optimization
  12. Exploiting term sparsity in the mSOS
  13. Graphs and sparse positive semidefinite matrices
  14. Support of a polynomial
  15. Term sparsity procedure
  16. ...and 12 more sections

Key Result

Theorem 1

If Assumption archimedian is satisfied, then $f_r \nearrow f^*$ as $r \rightarrow \infty$ in lassereSOS, i.e., there exists an $\tilde{r} \geq r$ such that

Figures (13)

  • Figure 1: Illustrative problem: boundary conditions and cross-section parametrization.
  • Figure 2: Feasible set of the illustration problem in terms of the cross-section areas $\bm{a}$. The red arrow, which is a normal vector of the triangular patch in the lower left corner, indicates the direction of minimization. Scattered green points denote local minimizers. The boundaries of the feasible set are drawn as semitransparent surfaces, and their color denotes the weight $w(\bm{a})$.
  • Figure 3: Feasible set of the first-order moments for (a) $r=1$ and (b) $r=2$. The red arrow, which is a normal vector of the triangular patch in the lower left corner, indicates the direction of minimization. The scattered red points denote the relaxation lower bounds and the yellow points the reconstructed upper bounds based on Section \ref{['sec:weightUB']}. The boundaries of the feasible set are drawn as solid, whereas the relaxation feasible set is drawn as a transparent surface, its color denoting the weight $w(\bm{a})$.
  • Figure 4: (a) Example of a non-chordal graph with two connected components, its (b) maximal and (c) minimal chordal extensions.
  • Figure 5: The TSP graph of the illustrative problem \ref{['POP-example']} with a relaxation degree $r=2$.
  • ...and 8 more figures

Theorems & Definitions (18)

  • Example 1
  • Definition 1: Moment and localizing matrix
  • Theorem 1: henrion2006convergent, Theorem 2.2
  • Example 2
  • Theorem 2
  • Example 3
  • Example 4
  • Proposition 1
  • Definition 2: Non-mixed term monomial basis
  • Remark 1
  • ...and 8 more