v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
472 CHAPTER 7. PROXIMITY PROBLEMS7.3.2 Minimization of affine dimension in Problem 3When the desired affine dimension ρ is diminished, Problem 3 (1184) isdifficult to solve [133,3] because the feasible set in R N(N−1)/2 loses convexity.By substituting rank envelope (1168) into Problem 3, then for any given Hwe get a convex problemminimize ‖D − H‖ 2 FDsubject to − tr(V DV ) ≤ κ ρ(1200)D ∈ EDM Nwhere κ ∈ R + is a constant determined by cut-and-try. Given κ , problem(1200) is a convex optimization having unique solution in any desiredaffine dimension ρ ; an approximation to Euclidean projection on thatnonconvex subset of the EDM cone containing EDMs with correspondingaffine dimension no greater than ρ .The SDP equivalent to (1200) does not move κ into the variables as onpage 462: for nonnegative symmetric input H and distance-square squaredvariable ∂ as in (1188),minimize − tr(V (∂ − 2H ◦D)V )∂ , D[ ]∂ij d ijsubject to≽ 0 , j > i = 1... N −1d ij 1− tr(V DV ) ≤ κ ρD ∈ EDM N(1201)∂ ∈ S N hThat means we will not see equivalence of this cenv(rank)-minimizationproblem to the non−rank-constrained problems (1187) and (1189) like wesaw for its counterpart (1170) in Problem 2.Another approach to affine dimension minimization is to project insteadon the polar EDM cone; discussed in6.8.1.5.
7.3. THIRD PREVALENT PROBLEM: 4737.3.3 Constrained affine dimension, Problem 3When one desires affine dimension diminished further below what can beachieved via cenv(rank)-minimization as in (1201), spectral projection can beconsidered a natural means in light of its successful application to projectionon a rank ρ subset of the positive semidefinite cone in7.1.4.Yet it is wrong here to zero eigenvalues of −V DV or −V GV or avariant to reduce affine dimension, because that particular method comesfrom projection on a positive semidefinite cone (1138); zeroing thoseeigenvalues here in Problem 3 would place an elbow in the projectionpath (confer Figure 109) thereby producing a result that is necessarilysuboptimal. Problem 3 is instead a projection on the EDM cone whoseassociated spectral cone is considerably different. (5.11.2.3) Proper choiceof spectral cone is demanded by diagonalization of that variable argument tothe objective:7.3.3.1 Cayley-Menger formWe use Cayley-Menger composition of the Euclidean distance matrix to solvea problem that is the same as Problem 3 (1184): (5.7.3.0.1)[ ] [ ]∥ minimize0 1T 0 1T ∥∥∥2D∥ −1 −D 1 −H[ ]F0 1T(1202)subject to rank≤ ρ + 21 −DD ∈ EDM Na projection of H on a generally nonconvex subset (when ρ < N −1) of theEuclidean distance matrix cone boundary rel∂EDM N ; id est, projectionfrom the EDM cone interior or exterior on a subset of its relative boundary(6.6, (982)).Rank of an optimal solution is intrinsically bounded above and below;[ ] 0 1T2 ≤ rank1 −D ⋆ ≤ ρ + 2 ≤ N + 1 (1203)Our proposed strategy ([ for low-rank ]) solution is projection on that subset0 1Tof a spectral cone λ1 −EDM N (5.11.2.3) corresponding to affine
- 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
- Page 431 and 432: 6.8. DUAL EDM CONE 4316.8.1.7 Schoe
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- Page 435 and 436: 6.10. POSTSCRIPT 435When D is an ED
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- Page 439 and 440: In contrast, order of projection on
- Page 441 and 442: 441HS N h0EDM NK = S N h ∩ R N×N
- Page 443 and 444: 4437.0.3 Problem approachProblems t
- Page 445 and 446: 7.1. FIRST PREVALENT PROBLEM: 445fi
- Page 447 and 448: 7.1. FIRST PREVALENT PROBLEM: 4477.
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- Page 451 and 452: 7.1. FIRST PREVALENT PROBLEM: 4517.
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- Page 455 and 456: 7.1. FIRST PREVALENT PROBLEM: 455Th
- Page 457 and 458: 7.2. SECOND PREVALENT PROBLEM: 457O
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- Page 463 and 464: 7.2. SECOND PREVALENT PROBLEM: 463c
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- Page 467 and 468: 7.3. THIRD PREVALENT PROBLEM: 467fo
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- Page 471: 7.3. THIRD PREVALENT PROBLEM: 4717.
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- Page 478 and 479: 478 CHAPTER 7. PROXIMITY PROBLEMSth
- Page 480 and 481: 480 APPENDIX A. LINEAR ALGEBRAA.1.1
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- Page 518 and 519: 518 APPENDIX B. SIMPLE MATRICESB.1
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472 CHAPTER 7. PROXIMITY PROBLEMS7.3.2 Minimization of affine dimension in Problem 3When the desired affine dimension ρ is diminished, Problem 3 (1184) isdifficult to solve [133,3] because the feasible set in R N(N−1)/2 loses convexity.By substituting rank envelope (1168) into Problem 3, then for any given Hwe get a convex problemminimize ‖D − H‖ 2 FDsubject to − tr(V DV ) ≤ κ ρ(1200)D ∈ EDM Nwhere κ ∈ R + is a constant determined by cut-and-try. Given κ , problem(1200) is a convex optimization having unique solution in any desiredaffine dimension ρ ; an approximation to Euclidean projection on thatnonconvex subset of the EDM cone containing EDMs with correspondingaffine dimension no greater than ρ .The SDP equivalent to (1200) does not move κ into the variables as onpage 462: for nonnegative symmetric input H and distance-square squaredvariable ∂ as in (1188),minimize − tr(V (∂ − 2H ◦D)V )∂ , D[ ]∂ij d ijsubject to≽ 0 , j > i = 1... N −1d ij 1− tr(V DV ) ≤ κ ρD ∈ EDM N(1201)∂ ∈ S N hThat means we will not see equivalence of this cenv(rank)-minimizationproblem to the non−rank-constrained problems (1187) and (1189) like wesaw for its counterpart (1170) in Problem 2.Another approach to affine dimension minimization is to project insteadon the polar EDM cone; discussed in6.8.1.5.
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