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** :

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

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

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

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