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On a probabilistic global optimizer derived from the Walker slice sampling

Aditya Gupta, Souvik Das, Debasish Chatterjee

TL;DR

This article presents a zeroth order probabilistic global optimization algorithm -- SwiftNav -- for (not necessarily convex) functions over a compact domain and demonstrates the effectiveness and accuracy of SwiftNav in high-dimensional benchmark optimization problems.

Abstract

This article presents a zeroth order probabilistic global optimization algorithm -- SwiftNav -- for (not necessarily convex) functions over a compact domain. A discretization procedure is deployed on the compact domain, starting with a small step-size $h > 0$ and subsequently adaptively refining it in the course of a simulated annealing routine utilizing the Walker slice and the Gibbs sampler, in order to identify a set of global optimizers up to good precision. SwiftNav is parallelizable, which helps with scalability as the dimension of decision variables increases. Several numerical experiments are included here to demonstrate the effectiveness and accuracy of SwiftNav in high-dimensional benchmark optimization problems.

On a probabilistic global optimizer derived from the Walker slice sampling

TL;DR

This article presents a zeroth order probabilistic global optimization algorithm -- SwiftNav -- for (not necessarily convex) functions over a compact domain and demonstrates the effectiveness and accuracy of SwiftNav in high-dimensional benchmark optimization problems.

Abstract

This article presents a zeroth order probabilistic global optimization algorithm -- SwiftNav -- for (not necessarily convex) functions over a compact domain. A discretization procedure is deployed on the compact domain, starting with a small step-size and subsequently adaptively refining it in the course of a simulated annealing routine utilizing the Walker slice and the Gibbs sampler, in order to identify a set of global optimizers up to good precision. SwiftNav is parallelizable, which helps with scalability as the dimension of decision variables increases. Several numerical experiments are included here to demonstrate the effectiveness and accuracy of SwiftNav in high-dimensional benchmark optimization problems.

Paper Structure

This paper contains 19 sections, 1 theorem, 16 equations, 14 figures, 8 tables, 3 algorithms.

Table of Contents

  1. Motivation and significance
  2. Problem description
  3. $\algoname$: Our algorithm
  4. Architecture
  5. Key functionalities
  6. Notation used in Algorithm \ref{['alg:swiftnav']}
  7. Illustrative experiments
  8. Ackley function
  9. Levy function
  10. Process synthesis MINLP
  11. Quadratically constrained quartic program
  12. Chebyshev center problem
  13. SDP - Truss1
  14. Conclusion
  15. An exposition to Walker slice sampling techniques
  16. ...and 4 more sections

Key Result

proposition 1

Consider the Walker slice sampler with the transition probability mass function $p_A \cprobof[]{\cdot \given \cdot }$ defined in eq:transition probability. Denote the normalized version of $\pi$ by where $Z>0$ is the normalization parameter. If $r_0 = \max\{1, s-k+1\}$, then $\sum_{r=r_0}^{s+k-1} p_A\cprobof[]{r\given s} = 1$. Moreover,

Figures (14)

  • Figure 1: Schematic of $\algoname$.
  • Figure 2: Plot of the Ackley function on $[-10, 10]^2$.
  • Figure 3: Plots showing the average log-scaled regret values against iterations for the Ackley function in dimension $10^3$. The curve shows the mean value with standard error represented as a band measured across 10 independent simulations. The complete numerical experiment run for 1000 simulations is on the left and a zoomed-in version showing the first 200 iterations out of 1000 is displayed on the right. Here too, the burn-in time is 20 iterations, which is excluded from the figures.
  • Figure 4: Plot depicting the log regret of the values obtained against iterations.
  • Figure 5: Plot depicting the log regret of the best values obtained thus far against iterations.
  • ...and 9 more figures

Theorems & Definitions (1)

  • proposition 1: ref:SGW-14