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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.
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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...

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