A Workout in Computational Finance

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 x⁰ ∈ S, x¹ ∈ 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, ...