By Richard B. Holmes (auth.)

ISBN-10: 3540057641

ISBN-13: 9783540057642

ISBN-10: 3540371826

ISBN-13: 9783540371823

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**Example text**

A ~lobal in X at f x o, minimum at in small then (2) and Xo, that (X, f). the set of all s o l u t i o n s set of which of a c o n v e x p r o g r a m is c l o s e d w h e n e v e r f is isc X. d) We are now ready for the second application of the lemma be a ics and i0 a). Theorem. (Pshenichnii, Rockafellar) Let X in is, 3i f ~ Conv either K. (X). dom Then Suppose that K is a convex set in (f) /-~int (K) + ~ xo s K and only if f is continuous Identifying ~f(Xo) such that at some point in is a solution of the convex program -N(Xo,K ) ~ Proof.

Definition. The normal cone to K at x the polar here of a cone at K 0 is (S (x o,K)-xo) o We are taking w*-closed see from convex at the d e f i n i t i o n s functionals That cone whose @ in involved real parts denoted that attain this so we obtain N(Xo,K). cone We easily consists their m a x i m u m over a K of those at x o. is, N ( X o , K ) = {~ ~ X*: In p a r t i c u l a r , when K ~, x o c K. empty X*, @, provided by definition. K re ~(Xo) is a subspace We e m p h a s i z e of = max X, that w h e n re ~(K)}.

Our interest We will have occasion convex sets to compute (the constraint sets in normal normal cones cones to to in optimization problems) The mapping f ~ Vf(Xo) in later sections. §i0. Subdifferential a) Let X be a real ics and is a linear mapping into X *. convex Formulas from the linear Now the subdifferential cone Conv (X) play a role analogous preserves the cone operations, of convex and this subsets ~f(Xo) of mapping. X* is supposed For this linear f, g ~ Conv analogy (Moreau, (f)f'] dot (g) to mapping combinations.

### A Course on Optimization and Best Approximation by Richard B. Holmes (auth.)

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