By Radu Ioan Bot
This booklet provides new achievements and leads to the speculation of conjugate duality for convex optimization difficulties. The perturbation method for attaching a twin challenge to a primal one makes the thing of a initial bankruptcy, the place additionally an outline of the classical generalized inside element regularity stipulations is given. A crucial function within the booklet is performed by way 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 broader applicability than the generalized inside aspect ones. The reader additionally gets deep insights into biconjugate calculus for convex services, the relatives among assorted latest robust duality notions, but additionally into a number of unconventional Fenchel duality issues. the ultimate a part of the ebook is consecrated to the functions of the convex duality concept within the box of monotone operators.
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Additional resources for Conjugate duality in convex optimization
Epi g / the image of the set R ! 2. f C g ı A/ D cl! D cl! R R Â epi A g / D cl! y / C f . 2) Proof. 9) give rise to the following characterization of the conjugate of f C g ı A (see also ). 3. 4. D A / sup f f . y /g ; y 2Y one can formulate the following closedness-type regularity condition (cf. RCiA /, i 2 f2; 20 ; 200 g. 3) guarantees the "-subdifferential formula for f C g ı A, fact which was also pointed out in [78, 127]. 5. Ax/ : 0 "1 ;"2 0 "1 C"2 D" Next, we take X D Y and assume that A is the identity operator on X , while f; g W X !
DfCFL i n / and the dual has an optimal solution. 9. Pf i n /, too. PfCi n / and its Fenchel dual it is enough to assume that (cf. PfCi n / and its Lagrange dual it is enough to assume that (cf. x 0 / < 0; i 2 N holds. 4 The Composed Convex Optimization Problem In this section, we employ the general approach described in the first section to the formulation of two conjugate duals to an unconstrained composed convex optimization problem. We also derive generalized interior point regularity conditions and provide strong duality theorems.
Z a proper, C -convex and C -epi closed function such that dom f \ S \ g 1 . C / ¤ ;. We define h W X ! dom f \ dom h/ \ . C / ¤ ;. The perturbation functions considered in Section 6 become ˆC C1 W X Z ! x; z/ and ˆC C2 W X X Z ! x/ 2 C g Both ˆC C1 and ˆC C2 are proper, convex and lower semicontinuous. It is worth mentioning that for guaranteeing the lower semicontinuity of the perturbation functions in this case, it is not necessary to assume that h is star C -lower semicontinuous. P C /. 7 furnish the following statement.
Conjugate duality in convex optimization by Radu Ioan Bot