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An Introduction to Optimization, 4th Edition by Stanislaw H. Zak, Edwin K. P. Chong

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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:

equation

where f : n →, h : nm, mn, and g: np. For the general problem above, we adopt the following definitions.

Definition 21.1 An inequality constraint gj(x) ≤ 0 is said to be active at x* if gj(x*) = 0. It is inactive at x* if gj(x*) < 0.

By convention, we consider an equality constraint hi(x) = 0 to be always active.

Definition 21.2 Let x* satisfy h(x*) = 0, g(x*) ≤ 0, and let J(x*) be the index set of active inequality ...

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