Adapting the centred simplex gradient to compensate for misaligned sample points

The centred simplex gradient (CSG) is a popular gradient approximation technique in derivative-free optimization. Its computation requires a perfectly symmetric set of sample points and is known to provide an accuracy of O(?(2)), where ? is the radius of the sampling set. In this paper, we consider the situation where the set of sample points is not perfectly symmetric. By adapting the formula for the CSG to compensate for the misaligned points, we define a new Adapted-CSG. We study the error bounds and the numerical stability of the Adapted-CSG. We also present numerical examples to demonstrate its properties relative to each new parameter and make a comparison to an alternative method.