By Radu Ioan Bot

This booklet offers new achievements and ends up in the speculation of conjugate duality for convex optimization difficulties. The perturbation method for attaching a twin challenge to a primal one makes the item of a initial bankruptcy, the place additionally an summary of the classical generalized inside element regularity stipulations is given. A crucial position within the e-book is performed by way of the formula of generalized Moreau-Rockafellar formulae and closedness-type stipulations, the latter constituting a brand new category of regularity stipulations, in lots of occasions with a much wider applicability than the generalized inside aspect ones. The reader additionally gets deep insights into biconjugate calculus for convex services, the kin among various latest powerful duality notions, but in addition into a number of unconventional Fenchel duality themes. the ultimate a part of the booklet is consecrated to the functions of the convex duality conception within the box of monotone operators.

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Additional resources for Conjugate duality in convex optimization

Example text

R, we have that ˆ is convex and, consequently, the infimal value function of ˆ , infy 2Y ˆ . ; y / W X ! R is also convex. For a subset of X and a function defined on X the closure and the lower semicontinuous hull, respectively, in the weak topology are denoted by cl! , while cl! X ; X // R. Here, we denote by R the natural topology on R. The following theorem can be obtained from [110] and plays a determinant role in the investigations we make in this chapter. 1. Let ˆ W X Y ! dom ˆ/. Then for each x 2 X one has  à inf ˆ .

F1 P (ii) epi. m i D1 fi / D cl! f1 Pm : : : fm // D cl! R. 15. We have ! 16. 9) i D1 i D1 † /. RCi† /, i 2 f2; 20 ; 200 g. 13 leads to the following result. 17. 9) is fulfilled if and only if for all x 2 X and " 0 one has ! 17 in connection to the so-called strong conical hull intersection property. x/: i D1 i D1 The notion of strong CHIP has been introduced by Deutsch, Li and Ward in [63] in Hilbert spaces and has proved to be useful when dealing with best approximation problems as well as in the conjugate duality theory (cf.

Inf ˆ . ; y / : y 2Y This concludes the proof. A first consequence of this result follows. 2. Let ˆ W X Y ! ˆ//. ˆ. ; 0// / D cl!  R  epi Ãà inf ˆ . ; y / y 2Y D cl! 2) Proof. x ; r/ 2 epi infy 2Y ˆ . ; y / . x ; r/ 2 epi infy 2Y ˆ . x ; y / Ä r C ". ˆ . x ; r/ 2 cl! epi ˆ //. epi ˆ /  epi  à inf ˆ . ; y /  cl! 2). ˆ. ; 0// / D cl! R epi infy 2Y ˆ . ; y / , we get the desired conclusion. The theorems proved above lead to the following statement, given also in [49] (see also [19]). 3. Let ˆ W X Y !

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