# CHAPTER 21

# PROBLEMS WITH INEQUALITY CONSTRAINTS

# 21.1 Karush-Kuhn-Tucker Condition

In Chapter 20 we analyzed constrained optimization problems involving only equality constraints. In this chapter we discuss extremum problems that also involve inequality constraints. The treatment in this chapter parallels that of Chapter 20. In particular, as we shall see, problems with inequality constraints can also be treated using Lagrange multipliers.

We consider the following problem:

where *f* : ^{n} →, ** h** :

^{n}→

^{m},

*m*≤

*n*, and

**:**

*g*^{n}→

^{p}. For the general problem above, we adopt the following definitions.

**Definition 21.1** An inequality constraint *g*_{j}(** x**) ≤ 0 is said to be

*active*at

*** if**

*x**g*

_{j}(

***) = 0. It is**

*x**inactive*at

*** if**

*x**g*

_{j}(

***) < 0.**

*x*By convention, we consider an equality constraint *h*_{i}(** x**) = 0 to be always active.

**Definition 21.2** Let ** x*** satisfy

**(**

*h****) =**

*x***0**,

**(**

*g****) ≤**

*x***0**, and let

*J*(

***) be the index set of active inequality ...**

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