By Radu Ioan Bot
This publication provides new achievements and ends up in the speculation 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 outline of the classical generalized inside element regularity stipulations is given. A imperative function within the booklet is performed through 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 aspect ones. The reader additionally gets deep insights into biconjugate calculus for convex services, the kinfolk among diverse present robust duality notions, but in addition into numerous unconventional Fenchel duality themes. the ultimate a part of the publication is consecrated to the purposes of the convex duality conception within the box of monotone operators.
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Extra resources for Conjugate duality in convex optimization
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. 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.
D CF / no ordering relation can be established (see also [36, 37]). P C / and the three duals treated above. To this end, we additionally assume that S Â X is a convex set, f W X ! R is a 24 I Perturbation Functions and Dual Problems convex function and g W X ! Z is a C -convex function. Under these hypotheses, the three perturbation functions are proper and convex and 0 is an element in the projection of their domains on the space of the perturbation variables. D CL /. RC1ˆ / states in this particular case that there exists x 0 2 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 . RC1ˆ / states that there exists x 0 2 dom f \ S \ g 1 . y; z/ 7! C /. dom f CFL /. y; z/ 2 dom f C 27 epi. C /. dom ˆ D dom f C epi. C /. RC2CFL / X and Z are Fr´echet spaces, S is closed, f is lower semicontinuous, g is C -epi closed and 0 2 sqri dom f C epi. C / . RC2C0 FL / X and Z are Fr´echet spaces, S is closed, f is lower semicontinuous, g is C -epi closed and 0 2 core dom f C epi.