By Radu Ioan Bot
This ebook offers new achievements and ends up in the speculation of conjugate duality for convex optimization difficulties. The perturbation strategy for attaching a twin challenge to a primal one makes the article of a initial bankruptcy, the place additionally an summary of the classical generalized inside element regularity stipulations is given. A principal position within the booklet is performed by means of the formula of generalized Moreau-Rockafellar formulae and closedness-type stipulations, the latter constituting a brand new classification of regularity stipulations, in lots of events with a much wider applicability than the generalized inside aspect ones. The reader additionally gets deep insights into biconjugate calculus for convex features, the kin among diverse current powerful duality notions, but additionally into a number of unconventional Fenchel duality themes. the ultimate a part of the ebook is consecrated to the purposes of the convex duality conception within the box of monotone operators.
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Extra resources for Conjugate duality in convex optimization
D C C2 / and the dual has an optimal solution. 3. z h/ . P C C /. P C C /. D /. D C C1 / strong duality holds, too. Chapter II Moreau–Rockafellar Formulae and Closedness-Type Regularity Conditions 5 Generalized Moreau–Rockafellar Formulae Throughout this chapter, we assume that all topological dual spaces of the separated locally convex spaces considered are endowed with the corresponding weak topologies. P G/. These will be particularized in the following sections to the different classes of convex functions and corresponding convex optimization problems, respectively, introduced in the previous chapter (see also ).
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 / Â epi Â Ã inf ˆ . ; y / Â 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  (see also ). 3. Let ˆ W X Y ! ˆ//. 3) As usual, we write instead of inf (sup) for an attained infimum (supremum) min (max). In the following, we introduce the concept of stable strong duality. x; 0/ x2X hx ; xig: The function ˆx W X Y !
Z g/ C ıS / D [z and this is a closed set. 3) via a "-subdifferential sum formula. Also here, it is worth noticing that for the result below no convexity or topological assumptions for the sets and functions involved are needed. 6. 3 (see also ). 7. 8. z g/S . X ; X // z 2C R. RCiCFL /; i 2 f2; 20 ; 200 g. RC4CFL /. RCfCiFL n / are not fulfilled, is obvious. Further, since dom f C epi C . RC2C00FL /) are not valid. z g/ C ıS / D f1g RC C [z 0 Œ0; z RC D Œ1; C1/ RC and this is a closed set.
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