Fmin

fmin is a name used for a class of numerical optimization routines that seek the minimum of a scalar function of several variables, typically without requiring gradient information. It is most commonly associated with the Nelder-Mead simplex method, a derivative-free algorithm that operates on a simplex of n+1 points in n-dimensional space.

The Nelder-Mead algorithm iteratively evaluates the objective function at the vertices of the current simplex and performs operations—reflection, expansion, contraction, and shrink—to move the simplex toward regions of lower function value. Convergence is determined by criteria such as small changes in the function values or vertex positions, or by reaching a maximum number of iterations. The method is robust to non-smooth or noisy functions and does not require the objective to be differentiable, but it does not guarantee convergence to a global minimum and its performance can be sensitive to scaling and the choice of initial point.

In practice, fmin appears in various libraries as a convenience wrapper for unconstrained minimization. It typically takes a user-supplied function to minimize, an initial guess for the decision variables, and optional parameters such as tolerance and maximum iterations. The routine returns the minimizer and the objective function value at that point, with additional output sometimes available depending on the implementation.

Limitations of fmin include potential inefficiency in high-dimensional problems and poor performance on highly multimodal landscapes. When derivatives are available, gradient-based methods or more advanced optimizers may offer faster convergence. For robust general use, modern practice often prefers interfaces that provide flexible method selection and clearer convergence diagnostics, such as a generic minimize function with a Nelder-Mead option.

See also: Nelder-Mead algorithm, Powell’s method, gradient-based optimization, constrained optimization, minimize interfaces.