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An approximate Itô-SDE based simulated annealing algorithm for multivariate design optimization problems

A. Batou

TL;DR

This paper tackles global optimization for high-dimensional, non-convex design problems governed by large computational models. It develops an approximate Itô stochastic differential equation (ISDE) based simulated annealing framework that replaces the Metropolis-Hastings sampler with an ISDE generator and uses a polyharmonic spline surrogate to cheaply evaluate and differentiate the cost function. The approach includes regularized constraint handling, adaptive step sizing, and surrogate enrichment to progressively improve the search surface, enabling efficient exploration of large design spaces. Demonstrations on Ackley’s function and a Finite Element stiffness-design example show improved convergence to near-global optima with reduced reliance on expensive function evaluations.

Abstract

This research concerns design optimization problems involving numerous design parameters and large computational models. These problems generally consist in non-convex constrained optimization problems in large and sometimes complex search spaces. The classical simulated annealing algorithm rapidly loses its efficiency in high search space dimension. In this paper a variant of the classical simulated annealing algorithm is constructed by incorporating (1) an Itô stochastic differential equation generator (ISDE) for the transition probability and (2) a polyharmonic splines interpolation of the cost function. The control points are selected iteratively during the research of the optimum. The proposed algorithm explores efficiently the design search space to find the global optimum of the cost function as the best control point. The algorithm is illustrated on two applications. The first application consists in a simple function in relatively high dimension. The second is related to a Finite Element model.

An approximate Itô-SDE based simulated annealing algorithm for multivariate design optimization problems

TL;DR

This paper tackles global optimization for high-dimensional, non-convex design problems governed by large computational models. It develops an approximate Itô stochastic differential equation (ISDE) based simulated annealing framework that replaces the Metropolis-Hastings sampler with an ISDE generator and uses a polyharmonic spline surrogate to cheaply evaluate and differentiate the cost function. The approach includes regularized constraint handling, adaptive step sizing, and surrogate enrichment to progressively improve the search surface, enabling efficient exploration of large design spaces. Demonstrations on Ackley’s function and a Finite Element stiffness-design example show improved convergence to near-global optima with reduced reliance on expensive function evaluations.

Abstract

This research concerns design optimization problems involving numerous design parameters and large computational models. These problems generally consist in non-convex constrained optimization problems in large and sometimes complex search spaces. The classical simulated annealing algorithm rapidly loses its efficiency in high search space dimension. In this paper a variant of the classical simulated annealing algorithm is constructed by incorporating (1) an Itô stochastic differential equation generator (ISDE) for the transition probability and (2) a polyharmonic splines interpolation of the cost function. The control points are selected iteratively during the research of the optimum. The proposed algorithm explores efficiently the design search space to find the global optimum of the cost function as the best control point. The algorithm is illustrated on two applications. The first application consists in a simple function in relatively high dimension. The second is related to a Finite Element model.

Paper Structure

This paper contains 14 sections, 34 equations, 19 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Design optimization problem and classical simulated annealing algorithm
  3. Construction of a new ISDE-based simulated annealing algorithm
  4. ISDE generator
  5. ISDE generator for random variable $\boldsymbol{A}_T$
  6. ISDE based simulated annealing algorithm
  7. Approximation of the cost function
  8. Approximate ISDE-based simulated annealing algorithm
  9. Application
  10. Ackley's test function
  11. Exact ISDE-based simulated annealing algorithm
  12. Approximate ISDE-based simulated annealing algorithm
  13. Finite Element model
  14. Conclusions

Figures (19)

  • Figure 1: Function $\boldsymbol{a} \mapsto \mathcal{D}(\boldsymbol{a})$ for $N=1$.
  • Figure 2: Function $\boldsymbol{a} \mapsto \mathcal{D}(\boldsymbol{a})$ for $N=2$.
  • Figure 3: Function $k \mapsto T_k$.
  • Figure 4: ISDE-based algorithm, $N=2$, functions $k \mapsto A_1^k$ and $k \mapsto A_2^k$.
  • Figure 5: ISDE-based algorithm, $N=200$, functions $k \mapsto A_i^k$ for $i=1,\ldots,200$.
  • ...and 14 more figures