Nonlinearly Constrained Optimization Problems

Abstarct

We again assume f, g ₁,…, g _m , h ₁,…,h _p to be continuously differentiable realvalued functions on ℝⁿ with n ∈ ℕ and m, p ∈ ℕ₀, and consider the problem

$$(P) \left\{ \begin{array}{lll}f(x)\longrightarrow \min \\ g_i(x)\leq 0 \quad\quad {\rm for} \; i \in \mathcal{I}: = \{1, \cdots, m\}\\ h_j(x) = 0 \quad\quad {\rm for} \; j \in \varepsilon: = \{1, \cdots, p\}\end{array}\right.$$

or short

$$(P) \left\{ \begin{array}{lll} f(x)\longrightarrow \min\\ x \in \mathcal{F}\end{array}\right.$$

with the set of feasible points

$$\mathcal{F}:=\left\{x \in \mathbb{R}^n | g_i(x)\leq 0 \; {\rm for} \; i \in \mathcal{I}, h_j(x) = 0 \; {\rm for} j \in \varepsilon \right\}.$$