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# 16 Optimization Techniques

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* ∈ ℝn that solves

is called an optimization problem for a given function f (x) : ℝn → 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 x0S, x1s, ... such that f(xk) → −∞ 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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