24. Unconstrained Minimization

Unconstrained minimization is used to minimize a function of many variables without any constraints on the variables, such as bounds. The methods available in TAO for solving these problems can be classified according to the amount of derivative information required:
    1. Function evaluation only -- Nelder-Mead method ( tao_nm)
    2. Function and gradient evaluations -- limited-memory, variable-metric method ( tao_lmvm) and nonlinear conjugate gradient method ( tao_cg)
    3. Function, gradient, and Hessian evaluations -- Newton line-search method ( tao_nls) and Newton trust-region method ( tao_ntr)
The best method to use depends on the particular problem being solved and the accuracy required in the solution. If a Hessian evaluation routine is available, then the Newton line-search and Newton trust-region methods will be the best performers. When a Hessian evaluation routine is not available, then the limited-memory, variable-metric method is likely to perform best. The Nelder-Mead method should be used only as a last resort when no gradient information is available.

Each solver has a set of options associated with it that can be set with command line arguments. A brief description of these algorithms and the associated options are discussed in this chapter.