v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
412 CHAPTER 6. EDM CONEdvec rel∂EDM 3dvec(11 T − E 3 )EDM N = cone{11 T − E N } = {t(11 T − E N ) | t ≥ 0} (1027)Figure 101: Three views of translated negated elliptope 11 T − E13(confer Figure 86) shrouded by truncated EDM cone. Fractal on EDMcone relative boundary is numerical artifact belonging to intersection withelliptope relative boundary. The fractal is trying to convey existence of aneighborhood about the origin where the translated elliptope boundary andEDM cone boundary intersect.
6.6. CORRESPONDENCE TO PSD CONE S N−1+ 4136.6.2.0.1 Expository. Define T E (11 T ) to be the tangent cone to theelliptope E at point 11 T ; id est,T E (11 T ) ∆ = {t(E − 11 T ) | t≥0} (1028)The normal cone K ⊥ E (11T ) to the elliptope at 11 T is a closed convex conedefined (E.10.3.2.1, Figure 129)K ⊥ E (11 T ) ∆ = {B | 〈B , Φ − 11 T 〉 ≤ 0, Φ∈ E } (1029)The polar cone of any set K is the closed convex cone (confer (258))K ◦ ∆ = {B | 〈B , A〉≤0, for all A∈ K} (1030)The normal cone is well known to be the polar of the tangent cone,and vice versa; [147,A.5.2.4]K ⊥ E (11 T ) = T E (11 T ) ◦ (1031)K ⊥ E (11 T ) ◦ = T E (11 T ) (1032)From Deza & Laurent [77, p.535] we have the EDM coneEDM = −T E (11 T ) (1033)The polar EDM cone is also expressible in terms of the elliptope. From(1031) we haveEDM ◦ = −K ⊥ E (11 T ) (1034)⋆In5.10.1 we proposed the expression for EDM DD = t11 T − E ∈ EDM N (901)where t∈ R + and E belongs to the parametrized elliptope E N t . We furtherpropose, for any particular t>0Proof. Pending.EDM N = cone{t11 T − E N t } (1035)Relationship of the translated negated elliptope with the EDM cone isillustrated in Figure 101. We speculateEDM N = limt→∞t11 T − E N t (1036)
- Page 361 and 362: 5.11. EDM INDEFINITENESS 361(confer
- Page 363 and 364: 5.11. EDM INDEFINITENESS 363we have
- Page 365 and 366: 5.11. EDM INDEFINITENESS 365For pre
- Page 367 and 368: 5.12. LIST RECONSTRUCTION 367where
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- Page 387 and 388: Chapter 6EDM coneFor N > 3, the con
- Page 389 and 390: 6.1. DEFINING EDM CONE 3896.1 Defin
- Page 391 and 392: 6.2. POLYHEDRAL BOUNDS 391This cone
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- Page 401 and 402: 6.5. EDM DEFINITION IN 11 T 401and
- Page 403 and 404: 6.5. EDM DEFINITION IN 11 T 403then
- Page 405 and 406: 6.5. EDM DEFINITION IN 11 T 4056.5.
- Page 407 and 408: 6.5. EDM DEFINITION IN 11 T 407D =
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- Page 419 and 420: 6.8. DUAL EDM CONE 419When the Fins
- Page 421 and 422: 6.8. DUAL EDM CONE 421Proof. First,
- Page 423 and 424: 6.8. DUAL EDM CONE 423EDM 2 = S 2 h
- Page 425 and 426: 6.8. DUAL EDM CONE 425whose veracit
- Page 427 and 428: 6.8. DUAL EDM CONE 4276.8.1.3.1 Exe
- Page 429 and 430: 6.8. DUAL EDM CONE 429has dual affi
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- Page 441 and 442: 441HS N h0EDM NK = S N h ∩ R N×N
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412 CHAPTER 6. EDM CONEdvec rel∂EDM 3dvec(11 T − E 3 )EDM N = cone{11 T − E N } = {t(11 T − E N ) | t ≥ 0} (1027)Figure 101: Three views of translated negated elliptope 11 T − E13(confer Figure 86) shrouded by truncated EDM cone. Fractal on EDMcone relative boundary is numerical artifact belonging to intersection withelliptope relative boundary. The fractal is trying to convey existence of aneighborhood about the origin where the translated elliptope boundary andEDM cone boundary intersect.
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