# Conjugate duality in convex optimization by Radu Ioan Bot

By Radu Ioan Bot

This ebook provides new achievements and leads to the idea of conjugate duality for convex optimization difficulties. The perturbation process 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 vital position within the publication 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 occasions with a much broader applicability than the generalized inside aspect ones. The reader additionally gets deep insights into biconjugate calculus for convex capabilities, the family among diverse current powerful duality notions, but additionally into a number of unconventional Fenchel duality themes. the ultimate a part of the ebook is consecrated to the functions of the convex duality thought within the box of monotone operators.

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D C C1 /. x 0 /. dom ˆC C1 //. dom f \ dom h/. dom ˆ Next, we show that whenever f and g are lower semicontinuous and h is star C lower semicontinuous, the lower semicontinuity of ˆC C1 is ensured. x; z/; which guarantees that ˆC C1 is lower semicontinuous. dom f \ dom h// ; which are in fact equivalent. dom f \ dom h// ¤ ;. RC2C0 C1 / have been introduced in [57] but under the assumption that h is a sequentially C -lower semicontinuous function. RC3C C1 / has been stated for the first time in [125].

Also here, it is worth noticing that for the result below no convexity or topological assumptions for the sets and functions involved are needed. 6. 3 (see also [24]). 7. 8. z g/S . X ; X // z 2C R. RCiCFL /; i 2 f2; 20 ; 200 g. RC4CFL /. RCfCiFL n / are not fulfilled, is obvious. Further, since dom f C epi C . RC2C00FL /) are not valid. z g/ C ıS / D f1g RC C [z 0 Œ0; z RC D Œ1; C1/ RC and this is a closed set. RC4CFL / is verified. 5) by means of a refined "subdifferential sum formula, without any convexity or topological assumption.

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 ! R and g W Y ! R and the linear continuous operator A W X ! dom f / \ dom g ¤ ;. Taking Z D Y , C D f0g and h W X ! x/ D Ax, we see that we are in a special case of the general setting from Section 6.