# SACOBRA with Online Whitening for Solving Optimization Problems with High Conditioning

## TL;DR

This paper introduces online whitening to enhance SACOBRA, a surrogate-assisted optimizer, enabling it to better handle high-conditioning problems by reducing early stagnation and improving accuracy in expensive optimization scenarios.

## Contribution

The paper proposes an online whitening technique for SACOBRA that addresses early stagnation and improves optimization accuracy on high-conditioning functions.

## Key findings

- Online whitening reduces optimization error by up to 1e12 compared to plain SACOBRA.
- SACOBRA with online whitening performs comparably to CMA-ES in long-term optimization when considering total evaluations.
- The approach is effective for problems with dimension D <= 20, especially in expensive evaluation settings.

## Abstract

Real-world optimization problems often have expensive objective functions in terms of cost and time. It is desirable to find near-optimal solutions with very few function evaluations. Surrogate-assisted optimizers tend to reduce the required number of function evaluations by replacing the real function with an efficient mathematical model built on few evaluated points. Problems with a high condition number are a challenge for many surrogate-assisted optimizers including SACOBRA. To address such problems we propose a new online whitening operating in the black-box optimization paradigm. We show on a set of high-conditioning functions that online whitening tackles SACOBRA's early stagnation issue and reduces the optimization error by a factor between 10 to 1e12 as compared to the plain SACOBRA, though it imposes many extra function evaluations. Covariance matrix adaptation evolution strategy (CMA-ES) has for very high numbers of function evaluations even lower errors, whereas SACOBRA performs better in the expensive setting (less than 1e03 function evaluations). If we count all parallelizable function evaluations (population evaluation in CMA-ES, online whitening in our approach) as one iteration, then both algorithms have comparable strength even on the long run. This holds for problems with dimension D <= 20.