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This is pag Printer: O Constrained Optimization The second part of this book is about minimizing functions subject to constraints on the variables. A general formulation for these problems is min x IR n f ( x )s u b j e c t t o ± c i ( x ) ± 0 , i E , c i ( x ) 0 , i I , (12.1) where f and the functions c i are all smooth, real-valued functions on a subset of IR n ,and I and E are two Fnite sets of indices. As before, we call f the objective function, while c i ,

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i E are the equality constraints and c i , i I are the inequality constraints . We defne the feasible set ± to be the set oF points x that satisFy the constraints; that is, ± ±{ x | c i ( x ) ± 0 , i E ; c i ( x ) 0 , i I } , (12.2) so that we can rewrite (12.1) more compactly as min x ± f ( x ) . (12.3) In this chapter we derive mathematical characterizations oF the solutions oF (12.3). As in the unconstrained case, we discuss optimality conditions oF two types. Necessary condi- tions are conditions that must be satisfed by any solution point (under certain assumptions). SufFcient conditions are those that, iF satisfed at a certain point x , guarantee that x is in Fact a solution. ±or the unconstrained optimization problem oF Chapter 2, the optimality conditions wereasFollows: Necessary conditions: Local unconstrained minimizers have f ( x ) ± 0and 2 f ( x ) positive semidefnite. SuFfcient conditions: Any point x at which f ( x ) ± 2 f ( x )ispos i t ive defnite is a strong local minimizer oF f . In this chapter, we derive analogous conditions to characterize the solutions oF constrained optimization problems. 2 1 EXAMPLES To introduce the basic principles behind the characterization oF solutions oF constrained optimization problems, we work through three simple examples. The discussion here is inFormal; the ideas introduced will be made rigorous in the sections that Follow. We start by noting one important item oF terminology that recurs throughout the rest oF the book.
Defnition 1. The active set A ( x ) at any feasible x consists of the equality constraint indices from E together with the indices of the inequality constraints i for which c i ( x ) ± 0 ; that is, A ( x ) ± E ∪{ i I | c i ( x ) ± 0 } . At a feasible point x , the inequality constraint i I is said to be active if c i ( x ) ± 0 and inactive if the strict inequality c i ( x ) > 0 is satisFed. A SINGLE EQUALITY CONSTRAINT E XAMPLE 1 Our Frst example is a two-variable problem with a single equality constraint: min x 1 + x 2 s.t. x 2 1 + x 2 2 2 ± 0 (12.9) (see ±igure 12.3). In the language of (12.1), we have f ( x ) ± x 1 + x 2 , I ±∅ , E ±{ 1 } ,and c 1 ( x ) ± x 2 1 + x 2 2 2. We can see by inspection that the feasible set for this problem is the circle of radius 2 centered at the origin—just the boundary of this circle, not its interior.

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notes1 - This is pag Printer O Constrained Optimization The...

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