By Radu Ioan Bot
This booklet provides new achievements and leads to the speculation of conjugate duality for convex optimization difficulties. The perturbation procedure 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 aspect regularity stipulations is given. A critical function within the booklet is performed through the formula of generalized Moreau-Rockafellar formulae and closedness-type stipulations, the latter constituting a brand new category 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 relatives among various current powerful duality notions, but additionally into a number of unconventional Fenchel duality subject matters. the ultimate a part of the publication is consecrated to the functions of the convex duality concept within the box of monotone operators.
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Extra info for Conjugate Duality in Convex Optimization (Lecture Notes in Economics and Mathematical Systems)
5) by means of a refined "subdifferential sum formula, without any convexity or topological assumption. 9. 10. y / y 2X A. 8. RCiCF /; i 2 f2; 20 ; 200 g. 11.
For a function g W X ! x/ 2 Zg and we say that g is proper when dom g ¤ ;. y/. The all x; y 2 X and all 2 Œ0; 1, g. x/ C C g. Also in this setting one has that g is C -convex if and only if epiC g is a convex set. z g/ W X ! x/i. z g/ D dom g. x/ 2 C g where S Â X is a given nonempty set and f W X ! R and g W X ! Z are proper functions fulfilling dom f \ S \ g 1 . C / ¤ ;. Here we denote by g 1 . x/ 2 C g. For the beginning, we take Z as the space of the perturbation variables and define ˆCL W X Z !
According to , we say that g W X ! z g/ is lower semicontinuous at x. The function g is said to be star C lower semicontinuous if it is star C -lower semicontinuous at all x 2 X . The notion of a C -epi closed function was introduced by Luc in . We call g C -epi closed if its C -epigraph epiC g is a closed set. 9] follows that every star C -lower semicontinuous function is C -epi closed. One can easily observe that when Z D R and C D RC the notions C -lower semicontinuity, and C -epi closedness coincide, as they collapse in this case into the classical lower semicontinuity.
Conjugate Duality in Convex Optimization (Lecture Notes in Economics and Mathematical Systems) by Radu Ioan Bot