By Radu Ioan Bot
This e-book provides new achievements and ends up in the idea of conjugate duality for convex optimization difficulties. The perturbation procedure 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 crucial function within the publication is performed by way of the formula of generalized Moreau-Rockafellar formulae and closedness-type stipulations, the latter constituting a brand new type 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 family among diversified present robust duality notions, but in addition into numerous unconventional Fenchel duality themes. the ultimate a part of the ebook is consecrated to the purposes of the convex duality thought within the box of monotone operators.
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Extra info for Conjugate duality in convex optimization
Z h/ . 8. f C g ı h/ D cl! D cl! z h/ . 6) Proof. 2). ˆC C2 / /. z h/ . x ; r/ 2 epi f C This leads to the desired result. 3 we obtain the following characterization for the conjugate of f C g ı h. 9. 10. z h/ . X ; X // R. RC3C C2 / fail. RC1C C2 / is not satisfied. x; x/ W x 2 Rg one has dom f dom g epiC h D [x2R .. dom f dom g epiC h/ if and only if [x2R .. 1; x xg is a closed linear subspace. RC2C C2 / is not fulfilled. RC2C00C2 /. z /2 ; C1/ D R RC and this is a closed set. RC4C C2 / is valid.
X1 ; x2 /T 2 R2 W 2x1 C x22 Ä 0 . Take f D ıU , g D idR and h D ıV . dom f \ dom h// D R, which is clearly a closed linear subspace. RC4C C1 / is valid, too. RC4C C2 / is satisfied or not in this situation. 1 yi Ä0 y1 ; 1/ D inf y1 Ä0 1 1 y1 y1 ; 1 y2 /g D D 0; but there is no y1 Ä 0 where this value is attained. RC4C C2 / is not valid. 44 II Moreau–Rockafellar Formulae and Closedness-Type Regularity Conditions 7 Stable Strong Duality for the Problem Having the Composition with a Linear Continuous Operator in the Objective Function Consider X and Y separated locally convex spaces, the proper, convex and lower semicontinuous functions f W X !
Z g/ C ıS / D cl! f R z 2C z 2C D cl! 3 has the following formulation (see also ). 3. 4. X ; X // R. 5. RC4CL / is, in fact, weaker than 52 II Moreau–Rockafellar Formulae and Closedness-Type Regularity Conditions these generalized interior point regularity conditions. RCfCiLn / are also not valid in this particular instance. 5. (cf. x/ D maxfx; 0g, x 2 R. RC1CL / fails, is obvious. RC2C00L /) fail, too. RC4CL / holds. z g/ C ıS / D [z and this is a closed set. 3) via a "-subdifferential sum formula.
Conjugate duality in convex optimization by Radu Ioan Bot