By Radu Ioan Bot
This booklet 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 item of a initial bankruptcy, the place additionally an outline of the classical generalized inside aspect regularity stipulations is given. A crucial function 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 type of regularity stipulations, in lots of events with a much wider applicability than the generalized inside element ones. The reader additionally gets deep insights into biconjugate calculus for convex capabilities, the kinfolk among diverse latest robust duality notions, but additionally into a number of unconventional Fenchel duality themes. the ultimate a part of the publication 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 (Lecture Notes in Economics and Mathematical Systems)
They can be derived from the ones given in Section 2 for Fenchel duality, as A is a nonempty convex set. A/ ¤ ;. Let us notice that for having A closed it is enough to assume that S is closed and g is C -epi closed. 2 we obtain the following result. 5. Let S Â X be a nonempty convex set, f W X ! R a proper and convex function and g W X ! Z a proper and C -convex function such that dom f \ S \ g 1 . C / ¤ ;. D CF / and the dual has an optimal solution. Next, we particularize the regularity conditions given in Section 1 by considering as perturbation function ˆCFL .
On the other hand, Ng and Song have given a sufficient condition for strong CHIP providing X is a Fr´echet space and a generalized interior point regularity condition is fulfilled (cf. 3]). 18 improves this result to general spaces and shows that it 50 II Moreau–Rockafellar Formulae and Closedness-Type Regularity Conditions works under a weaker regularity condition. For relations between strong CHIP and the so-called bounded linear regularity property which plays an important role when providing convergence rates for algorithms solving convex optimization problems, we refer to .
X/: i D1 i D1 The notion of strong CHIP has been introduced by Deutsch, Li and Ward in  in Hilbert spaces and has proved to be useful when dealing with best approximation problems as well as in the conjugate duality theory (cf. [4, 61–63]). Obviously, fC1 ; : : : ; Cm g has the strong CHIP if and only if ! 17 the following result. 18. X ; X // i D1 Ci ¤ ;. If has the strong CHIP. 1] for X a Banach space. 1] was provided in  for X an Euclidean space and under different supplementary geometric properties for the two convex and closed sets.