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Solution Polishing via Path Relinking for Continuous Black-Box Optimization

Dimitri Papageorgiou, Jan Kronqvist, Asha Ramanujam, James Kor, Youngdae Kim, Can Li

TL;DR

Two novel methods for performing solution polishing along one-dimensional curves rather than along straight lines are explored and it is shown that solution polishing along a curve is competitive with solution polishing using a state-of-the-art BBO solver.

Abstract

When faced with a limited budget of function evaluations, state-of-the-art black-box optimization (BBO) solvers struggle to obtain globally, or sometimes even locally, optimal solutions. In such cases, one may pursue solution polishing, i.e., a computational method to improve (or ``polish'') an incumbent solution, typically via some sort of evolutionary algorithm involving two or more solutions. While solution polishing in ``white-box'' optimization has existed for years, relatively little has been published regarding its application in costly-to-evaluate BBO. To fill this void, we explore two novel methods for performing solution polishing along one-dimensional curves rather than along straight lines. We introduce a convex quadratic program that can generate promising curves through multiple elite solutions, i.e., via path relinking, or around a single elite solution. In comparing four solution polishing techniques for continuous BBO, we show that solution polishing along a curve is competitive with solution polishing using a state-of-the-art BBO solver.

Solution Polishing via Path Relinking for Continuous Black-Box Optimization

TL;DR

Two novel methods for performing solution polishing along one-dimensional curves rather than along straight lines are explored and it is shown that solution polishing along a curve is competitive with solution polishing using a state-of-the-art BBO solver.

Abstract

When faced with a limited budget of function evaluations, state-of-the-art black-box optimization (BBO) solvers struggle to obtain globally, or sometimes even locally, optimal solutions. In such cases, one may pursue solution polishing, i.e., a computational method to improve (or ``polish'') an incumbent solution, typically via some sort of evolutionary algorithm involving two or more solutions. While solution polishing in ``white-box'' optimization has existed for years, relatively little has been published regarding its application in costly-to-evaluate BBO. To fill this void, we explore two novel methods for performing solution polishing along one-dimensional curves rather than along straight lines. We introduce a convex quadratic program that can generate promising curves through multiple elite solutions, i.e., via path relinking, or around a single elite solution. In comparing four solution polishing techniques for continuous BBO, we show that solution polishing along a curve is competitive with solution polishing using a state-of-the-art BBO solver.
Paper Structure (16 sections, 1 theorem, 5 equations, 26 figures, 2 tables, 4 algorithms)

This paper contains 16 sections, 1 theorem, 5 equations, 26 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Literature review
  3. Path relinking
  4. Line Search
  5. Methods
  6. Optimizing curves
  7. Curve search algorithm
  8. Numerical Experiments
  9. Experimental set up
  10. Test suite of functions
  11. Black-box optimization solvers compared
  12. Solution polishing methods compared
  13. Numerical Results
  14. Conclusions and future research directions
  15. Michalewicz function: Optimal objective function values
  16. ...and 1 more sections

Key Result

Lemma 1

Given a $D$-dimensional function $f$ that is smooth over $\mathcal{F}\subset\mathbb{R}^D$, the function is also smooth along any smooth curve that lies within $\mathcal{F}$.

Figures (26)

  • Figure 1: Orthogonal vs. straight path relinking.
  • Figure 2: Illustration of the discrete points on smooth curves obtained by solving \ref{['eq:QP']} using a different number of time steps. Observe that the coarse grid is only used for illustration purposes; for solution polishing, we use a significantly finer grid for the curve.
  • Figure 3: Examples of curves generated for the 4D Boha instance. Only dimensions 2, 3, and 4 are shown.
  • Figure 4: LineWalker's line search performance on the 4D Boha instances in Figure \ref{['fig:example_curves']}. The insets reveal that multiple improving solutions were found for both methods.
  • Figure 5: Gallery of test functions shown in two dimensions only. For some functions, the lower and upper bounds have been modified from those used in our computational experiments (see Table \ref{['table:test_fnc_table']}) to improve the visualization.
  • ...and 21 more figures

Theorems & Definitions (2)

  • Lemma 1
  • proof