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 x0 ∈ S, x1 ∈ s, ... 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
On some occasions, one may only find an x* satisfying
where is an open sphere with radius and midpoint x*. In this case, ...