12 The First-Order Optimality Condition

Optimisation

For a function $g(w)$ of $N$-dimensional independent variables $w\in {\mathbb{R}}^N$, the optimisation problem is :

$$\arg \underset{w}{\min }g(w)$$

Local Minimum Condition : A $w^{\ast }$ is the local minimum if it satisfies the first-order necessary condition for optimality.

$${\nabla }_{w}g(w^{\ast })=0_{N\times 1}$$

Stationary Point : The point satisfying the condition is called a stationary point. It could be a minimum, maximum or a saddle point.

The first order optimality condition :

The equation of first-order necessary condition can be written as a system of $N$ first-order equations:

$$\begin{array}{rl}\frac{\partial }{\partial w_1}g(w^{\ast })& =0\\ \frac{\partial }{\partial w_2}g(w^{\ast })& =0\\ ⋮& \\ \frac{\partial }{\partial w_N}g(w^{\ast })& =0\end{array}$$