Global optimization test problems based on random field composition

Abstract

The development and identification of effective optimization algorithms for non-convex real-world problems is a challenge in global optimization. Because theoretical performance analysis is difficult, and problems based on models of real-world systems are often computationally expensive, several artificial performance test problems and test function generators have been proposed for empirical comparative assessment and analysis of metaheuristic optimization algorithms. These test problems however often lack the complex function structures and forthcoming difficulties that can appear in real-world problems. This communication presents a method to systematically build test problems with various types and degrees of difficulty. By weighted composition of parameterized random fields, challenging test functions with tunable function features such as, variance contribution distribution, interaction order, and nonlinearity can be constructed. The method is described, and its applicability to optimization performance analysis is described by means of a few basic examples. The method aims to set a step forward in the systematic generation of global optimization test problems, which could lead to a better understanding of the performance of optimization algorithms on problem types with particular characteristics. On request an introductive MATLAB implementation of a test function generator based on the presented method is available.

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Notes

  1. 1.

    When “difficulty” or “hardness” is averaged over all possible search or optimization algorithms no problems are intrinsically harder than others [25]. However, for a particular optimization algorithm, some problem classes can be more difficult than others.

  2. 2.

    Such a passive map would be very memory intensive since the required memory scales with the number of elements \(m=\mathop \prod \nolimits _{d=1}^n r_d\) or \(m=r^n\) for a uniform resolution r and field dimension n. which already becomes problematic at modest resolutions and problem dimensions. A discrete field array of resolution \(r=10\) and dimension \(n=12\) would already require 8 TB (terabyte) of memory when each element takes 8 bit of storage.

  3. 3.

    Although the above statement is true for all \(p > 0\) the author recommends to use as a rule of thumb \(p \ge 1/d\), since for very small values of p the smoothness vanishes.

  4. 4.

    Explanation to the multi-index notation: The expression \(\mathop \sum \nolimits _{1\le i<j\le n} f_{i,j} ( {x_i,x_j })\) indicates a sum over all function decomposition terms with two variables for which \(1\le i<j\le n\). This applies similarly to all pairs of higher order interactions \(f_{i,j,\ldots ,n} \).

  5. 5.

    The variance for the terms expression 6, w.r.t. the corresponding sub domain in the unit hypercube can be expressed as: \(\text{ Var }( {f_{i,j,\ldots ,n} ({x_1,x_2,\ldots ,x_n })}) = \int f^2_{i,j,\ldots ,n} ( {x_1,x_2,\ldots ,x_n })\text{ d }x_i \ldots \text{ d }x_n \).

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Acknowledgments

The presented method is a generalized derivative of strategies developed for efficient optimization on structural simulations in vehicle design in the scope of the GRESIMO Project Funded by the European Commission under the 7th framework program under Grant Agreement 290050.

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Correspondence to Ramses Sala.

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Sala, R., Baldanzini, N. & Pierini, M. Global optimization test problems based on random field composition. Optim Lett 11, 699–713 (2017). https://doi.org/10.1007/s11590-016-1037-1

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Keywords

  • Global optimization
  • Metaheuristics
  • Random fields
  • Variable interactions
  • Performance test function
  • Artificial landscapes
  • Problem features
  • Optimization test problem