Read e-book online Conjugate duality in convex optimization PDF

By Radu Ioan Bot

This e-book offers new achievements and leads to the idea of conjugate duality for convex optimization difficulties. The perturbation method 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 element regularity stipulations is given. A valuable position within the e-book 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 broader 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 additionally into numerous unconventional Fenchel duality subject matters. the ultimate a part of the ebook is consecrated to the purposes of the convex duality concept within the box of monotone operators.

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P C /. D CFL /. 2. D CF /. 3 The Problem with Geometric and Cone Constraints 23 Proof. (i) Let z 2 C be fixed. z g/S . z g/S . D CL / follows automatically. (ii) Let y 2 X be fixed. z g/S . z g/S . D CF /. 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.

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. dom f \ S \ dom g/ C C and in order to guarantee the lower semicontinuity for ˆCL it is enough to assume that S is closed, f is lower semicontinuous and g is C -epi closed.

D CFL / and the dual has an optimal solution. 7. D CFL /. D CL /. D CF / strong duality holds, too. P C / for which we give some weak regularity conditions guaranteeing strong duality for the primal problem and the three conjugate dual problems assigned to it. Consider n n X D Rn , Z D R m , C D R m C , S Â R a nonempty convex set, f W R ! R a proper and convex function and g W Rn ! e. gi is convex for i D 1; : : : ; m, such that dom f \ S \ g 1 . Rm C / ¤ ;. x/ 0g where by “ ” we denote the partial order induced by the non-negative orthant Rm C on Rm .

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