Constrained Optimization
Constrained Optimization $$\min_{x \in R^n} f(x)$$
subject to: $$c_i (x) = 0, i \in E \text{(equality constraints)}$$ $$c_i (x) \geq 0, i \in I \text{(inequality constraints)}$$
Feasible Set $\Omega = { x_i | c_i(x) = 0, i \in E \text{ and } c_i(x) \geq 0, i \in I}$.
Case 1 $\min_{x \in R^n} f(x)$ subject to $c_1 (x) = 0$
x is local minimum if x + s $\notin \Omega$ or f(x+s) $\geq$ f(x).……