GNU Scientific Library – Reference Manual: Minimization Overview

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### 34.1 Overview

The minimization algorithms begin with a bounded region known to contain a minimum. The region is described by a lower bound a and an upper bound b, with an estimate of the location of the minimum x.

The value of the function at x must be less than the value of the function at the ends of the interval, This condition guarantees that a minimum is contained somewhere within the interval. On each iteration a new point x' is selected using one of the available algorithms. If the new point is a better estimate of the minimum, i.e. where f(x') < f(x), then the current estimate of the minimum x is updated. The new point also allows the size of the bounded interval to be reduced, by choosing the most compact set of points which satisfies the constraint f(a) > f(x) < f(b). The interval is reduced until it encloses the true minimum to a desired tolerance. This provides a best estimate of the location of the minimum and a rigorous error estimate.

Several bracketing algorithms are available within a single framework. The user provides a high-level driver for the algorithm, and the library provides the individual functions necessary for each of the steps. There are three main phases of the iteration. The steps are,

- initialize minimizer state, s, for algorithm T
- update s using the iteration T
- test s for convergence, and repeat iteration if necessary

The state for the minimizers is held in a `gsl_min_fminimizer` struct. The updating procedure uses only function evaluations (not derivatives).