A Novel Graphic Bending Transformation on Benchmark

TL;DR

This paper introduces a novel graphic bending transformation inspired by image processing to deform benchmark functions, revealing new challenges for optimizers and providing insights into algorithm sensitivity.

Contribution

The paper proposes a conformal bending transformation that alters the shape of benchmark functions without changing their fundamental properties, offering a new way to evaluate optimization algorithms.

Findings

Optimizers require more search effort on bent functions than rotated ones.

Bending affects the difficulty of optimization problems.

Parameter analysis shows sensitivity of algorithms to shape deformation.

Abstract

Classical benchmark problems utilize multiple transformation techniques to increase optimization difficulty, e.g., shift for anti centering effect and rotation for anti dimension sensitivity. Despite testing the transformation invariance, however, such operations do not really change the landscape's "shape", but rather than change the "view point". For instance, after rotated, ill conditional problems are turned around in terms of orientation but still keep proportional components, which, to some extent, does not create much obstacle in optimization. In this paper, inspired from image processing, we investigate a novel graphic conformal mapping transformation on benchmark problems to deform the function shape. The bending operation does not alter the function basic properties, e.g., a unimodal function can almost maintain its unimodality after bent, but can modify the shape of interested area in the search space. Experiments indicate the same optimizer spends more search budget and encounter more failures on the conformal bent functions than the rotated version. Several parameters of the proposed function are also analyzed to reveal performance sensitivity of the evolutionary algorithms.