Powell Method

The Powell method in its basic form can be viewed as a gradient-free minimization algorithm. It requires repeated line search minimizations, which may be carried out using univariate gradient free , or gradient based procedures . It was introduced by M.J.D. Powell [1]. The procedure is described in the algorithm steps below.

The minimization problem considered is:

1. Initialization

Select an accuracy ε > 0, and a starting point x ⁽⁰⁾. Set the initial search directions s ⁽ⁱ⁾ to be the unit vectors along each coordinate axis, for i = 1,..., n. Set the main iteration counter to k = 0, and the cycle counter i = 1. Initialize z ⁽¹⁾ = x ⁽⁰⁾. Set counter j = 0 for the case where step 2.2a is used.

2. Directional univariate minimization