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By Mirjam Dur

ISBN-10: 3826561155

ISBN-13: 9783826561153

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47 . c. problems is now specialized for convex maximization problems as follows. e. F = cl int F , ii there exists x0 2 D; t0 2 IR such that f x0 , t0 0 and t0 t. 1. 2 Let D 6= ;. x; t 2 Argmaxft : x; t 2 F g  maxff x , t : x; t 2 D  IR; t  t g = 0: 22 Chapter 3. Optimality Conditions for Convex Maximization Proof. =: Assume that there is ~x; t~ 2 D  IR such that t~  t and f ~x , t~ 0. e. 5. Then there exists ~x; t~ 2 D  IR with f ~x , t~  0 and t~ t, and hence f ~x , t 0.

1 gx; y := xy + x2 + y2 is a convex function. Proof. The Hessian H of g is  H = 21 12 ! which is a positive de nite matrix. c. decomposition h i h i xy = xy + x2 + y2 , x2 + y 2 : Polynomials of the form f x; y  = xny n , with n  2 even, also permit an elegant decomposition. We have h i h i xn yn = xnyn + x2n + y2n , x2n + y2n for n  2 even. This is entailed by the following proposition. 2 If n  2 is even, then gx; y := xnyn + x2n + y2n is a convex function. 44 Chapter 5. C. Decompositions Proof.

1 Convex Envelopes We begin with recalling the concept of the convex envelope of a nonconvex function which is a basic tool in theory and algorithms of nonconvex global optimization see Horst Tuy 51 or Horst et al. 47 and references therein. 55 56 Chapter 8. 1 Let C IRn be nonempty, compact and convex, and let f : C !  on C . Then the function 'C;f : C ! IR, 'C;f x := supfhx : h : C ! IR convex, h  f on C g is said to be the convex envelope of f over C . Notice that it is often convenient to eliminate formally the set C by setting 8 f x = : f x; x 2 C +1; x 2= C and replacing 'C;f accordingly by its extension 'C;f : IRn !

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Duality in Global Optimization: Optimality Conditions and Algorithmical Aspects by Mirjam Dur

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