By Radu Ioan Bot
This publication offers 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 summary of the classical generalized inside aspect regularity stipulations is given. A critical function within the publication 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 broader applicability than the generalized inside aspect ones. The reader additionally gets deep insights into biconjugate calculus for convex capabilities, the relatives among diversified latest robust duality notions, but additionally into numerous unconventional Fenchel duality issues. the ultimate a part of the e-book is consecrated to the functions of the convex duality thought within the box of monotone operators.
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Extra info for Conjugate duality in convex optimization
DG x /. 3 in ). 5. Let ˆ W X Y ! dom ˆ/. x ; y / 8x 2 X : y 2Y x2X Next, we derive by means of the considerations made above closedness-type regularity conditions and corresponding stable strong duality results for the primal–dual pairs treated in Sections 2–4. We provide also examples where the closedness-type conditions are satisfied, but the generalized interior point ones fail. 6 Stable Strong Duality for the Composed Convex Optimization Problem In this section, we work in the same setting as in the Section 4 and consider X and Z separated locally convex spaces, where Z is partially ordered by the nonempty convex cone C Â Z, f W X !
2]. Nevertheless, this function fails to be C -convex. The function in the example below is both C -convex and C -epi closed, but not star C -lower semicontinuous. 3. Consider the function ( 2 2 g W R ! x/ D C . x1 ; x/; if x > 0; 1R2 ; otherwise. C 3 The Problem with Geometric and Cone Constraints 25 It is easy to verify that g is R2C -convex and R2C -epi-closed, but not star R2C -lower semicontinuous. x/ D x; if x > 0; C1; otherwise, which fails to be lower semicontinuous. We come now to the class of regularity conditions which assume that X and Z are Fr´echet spaces.
R a proper, convex and lower semicontinuous function and g W X ! 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.
Conjugate duality in convex optimization by Radu Ioan Bot