Karush-Kuhn-Tucker Theory

Abstract

In the current chapter, we study the problem of minimizing a real-valued function $f(\boldsymbol{x})$ subject to the constraints

$$\displaystyle\begin{array}{rcl} g_{i}(\boldsymbol{x})& =& 0,\quad \quad 1 \leq i \leq p \\ h_{j}(\boldsymbol{x})& \leq & 0,\quad \quad 1 \leq j \leq q.\end{array}$$

All of these functions share some open set U ⊂R ⁿ as their domain. Maximizing $f(\boldsymbol{x})$ is equivalent to minimizing $-f(\boldsymbol{x})$, so there is no loss of generality in considering minimization. The function $f(\boldsymbol{x})$ is called the objective function, the functions $g_{i}(\boldsymbol{x})$ are called equality constraints, and the functions $h_{j}(\boldsymbol{x})$ are called inequality constraints. Any point $\boldsymbol{x} \in U$ satisfying all of the constraints is said to be feasible.