Efficient numerical optimization techniques play a key role in many of the calculation tasks in modern financial mathematics. Optimization problems occur for example in asset allocation, risk management and model calibration problems.

From a more theoretical point of view, the problem of finding an **x*** ∈ ℝ^{n} that solves

is called an optimization problem for a given function *f* (**x**) : ℝ^{n} → ℝ. The function *f* is called the objective function and *s* is the feasible region. When *s* is empty, the problem is called infeasible. If it is possible to find a sequence **x**^{0} ∈ *S,* **x**^{1} ∈ *s*, ... such that *f*(**x**^{k}) → −∞ for *k →* ∞, then the problem is called unbounded. For problems that are neither infeasible nor unbounded it may be possible to find a solution **x*** satisfying

Such an **x*** is called a global minimizer of the optimization problem (16.1) (see Figure 16.1).^{1}

On some occasions, one may only find an **x*** satisfying

where is an open sphere with radius and midpoint **x***. In this case, ...

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