v2009.01.01 - Convex Optimization

498 CHAPTER 7. PROXIMITY PROBLEMS
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✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ 0 ✟✟✟✟✟✟✟✟✟✟✟
EDM N
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❜ S N ❝ ❝❝❝❝❝❝❝❝❝❝❝
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❜ K = S N .
h ∩ R N×N
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R N×N
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Figure 128: Pseudo-Venn diagram: The EDM cone belongs to the
intersection of the symmetric hollow subspace with the nonnegative orthant;
EDM N ⊆ K (797). EDM N cannot exist outside S N h , but R N×N
+ does.
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7.0.1.2 Egregious input error under nonnegativity demand
More pertinent to the optimization problems presented herein where
C ∆ = EDM N ⊆ K = S N h ∩ R N×N
+ (1210)
then should some particular realization of a proximity problem demand
input H be nonnegative, and were we only to zero negative entries of a
nonsymmetric nonhollow input H prior to optimization, then the ensuing
projection on EDM N would be guaranteed incorrect (out of order).
Now comes a surprising fact: Even were we to correctly follow the
order-of-projection rule and provide H ∈ K prior to optimization, then the
ensuing projection on EDM N will be incorrect whenever input H has negative
entries and some proximity problem demands nonnegative input H .