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Optimal design of frame structures with mixed categorical and continuous design variables using the Gumbel-Softmax method

Mehran Ebrahimi, Hyunmin Cheong, Pradeep Kumar Jayaraman, Farhad Javid

TL;DR

The paper tackles optimizing frame structures with mixed categorical and continuous design variables using a gradient-based approach. It introduces the Gumbel-Softmax (GSM) method to render categorical sampling differentiable, enabling simultaneous optimization of both variable types via the GSMO framework and its bilevel variant BiGSMO. Through three large-scale case studies, GSMO/BiGSMO achieve substantial computational savings and competitive or superior objective values compared with gradient-free GA, while maintaining solution consistency. The work demonstrates the practical impact of differentiable sampling for scalable structural optimization, with potential extensions to nonlinear regimes and broader problem classes.

Abstract

In optimizing real-world structures, due to fabrication or budgetary restraints, the design variables may be restricted to a set of standard engineering choices. Such variables, commonly called categorical variables, are discrete and unordered in essence, precluding the utilization of gradient-based optimizers for the problems containing them. In this paper, incorporating the Gumbel-Softmax (GSM) method, we propose a new gradient-based optimizer for handling such variables in the optimal design of large-scale frame structures. The GSM method provides a means to draw differentiable samples from categorical distributions, thereby enabling sensitivity analysis for the variables generated from such distributions. The sensitivity information can greatly reduce the computational cost of traversing high-dimensional and discrete design spaces in comparison to employing gradient-free optimization methods. In addition, since the developed optimizer is gradient-based, it can naturally handle the simultaneous optimization of categorical and continuous design variables. Through three numerical case studies, different aspects of the proposed optimizer are studied and its advantages over population-based optimizers, specifically a genetic algorithm, are demonstrated.

Optimal design of frame structures with mixed categorical and continuous design variables using the Gumbel-Softmax method

TL;DR

The paper tackles optimizing frame structures with mixed categorical and continuous design variables using a gradient-based approach. It introduces the Gumbel-Softmax (GSM) method to render categorical sampling differentiable, enabling simultaneous optimization of both variable types via the GSMO framework and its bilevel variant BiGSMO. Through three large-scale case studies, GSMO/BiGSMO achieve substantial computational savings and competitive or superior objective values compared with gradient-free GA, while maintaining solution consistency. The work demonstrates the practical impact of differentiable sampling for scalable structural optimization, with potential extensions to nonlinear regimes and broader problem classes.

Abstract

In optimizing real-world structures, due to fabrication or budgetary restraints, the design variables may be restricted to a set of standard engineering choices. Such variables, commonly called categorical variables, are discrete and unordered in essence, precluding the utilization of gradient-based optimizers for the problems containing them. In this paper, incorporating the Gumbel-Softmax (GSM) method, we propose a new gradient-based optimizer for handling such variables in the optimal design of large-scale frame structures. The GSM method provides a means to draw differentiable samples from categorical distributions, thereby enabling sensitivity analysis for the variables generated from such distributions. The sensitivity information can greatly reduce the computational cost of traversing high-dimensional and discrete design spaces in comparison to employing gradient-free optimization methods. In addition, since the developed optimizer is gradient-based, it can naturally handle the simultaneous optimization of categorical and continuous design variables. Through three numerical case studies, different aspects of the proposed optimizer are studied and its advantages over population-based optimizers, specifically a genetic algorithm, are demonstrated.
Paper Structure (13 sections, 1 theorem, 30 equations, 9 figures, 5 tables, 4 algorithms)

This paper contains 13 sections, 1 theorem, 30 equations, 9 figures, 5 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Differentiable sampling via the GSM method
  4. The GSM method
  5. Optimization methodology
  6. Sensitivity analysis for continuous variables
  7. Sensitivity analysis for categorical variables
  8. Optimization scheme
  9. Case studies
  10. 72-bar truss structure
  11. 812-bar lattice structure
  12. 258-bar bridge structure
  13. Conclusion

Key Result

Proposition 1

The samples generated through eq:gsm have the same probability distribution as $\boldsymbol{\theta}$.

Figures (9)

  • Figure 1: The 72-bar truss structure. The typical story is repeated 4 times along the Z axis. The node and element numbering also follows suit.
  • Figure 2: Convergence plot of the 10 GSMO and GA runs for the truss structure
  • Figure 3: Geometry of the lattice structure
  • Figure 4: Available cross-sectional choices for the beams in the lattice structure
  • Figure 5: Optimum solutions of the lattice structure due to GSMO and BiGSMO
  • ...and 4 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof