v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
658 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS(Ky Fan, 1949) For B ∈ S N whose eigenvalues λ(B)∈ R N are arrangedin nonincreasing order, and for 1≤k ≤N [11,4.1] [215] [202,4.3.18][359,2] [239,2.1] [240]N∑i=N−k+1λ(B) i = infU∈ R N×kU T U=Itr(UU T B) = minimizeX∈ S N +tr(XB)subject to X ≼ ItrX = k(a)k∑λ(B) i = supi=1U∈ R N×kU T U=I= maximize (k − N)µ + tr(B − Z)µ∈R , Z∈S N +subject to µI + Z ≽ Btr(UU T B) = maximizeX∈ S N +tr(XB)subject to X ≼ ItrX = k(b)(c)= minimize kµ + trZµ∈R , Z∈S N +subject to µI + Z ≽ B(d)(1700)Given ordered diagonalization B = QΛQ T , (A.5.1) then an optimalU for the infimum is U ⋆ = Q(:, N − k+1:N)∈ R N×k whereasU ⋆ = Q(:, 1:k)∈ R N×k for the supremum is more reliably computed.In both cases, X ⋆ = U ⋆ U ⋆T . Optimization (a) searches the convex hullof outer product UU T of all N ×k orthonormal matrices. (2.3.2.0.1)For B ∈ S N whose eigenvalues λ(B)∈ R N are arranged in nonincreasingorder, and for diagonal matrix Υ∈ S k whose diagonal entries arearranged in nonincreasing order where 1≤k ≤N , we utilize themain-diagonal δ operator’s self-adjointness property (1418): [12,4.2]k∑Υ ii λ(B) N−i+1 = inf tr(ΥU T BU) = inf δ(Υ) T δ(U T BU)i=1U∈ R N×kU∈ R N×kU T U=I= minimizeV i ∈S Ni=1U T U=I()∑tr B k (Υ ii −Υ i+1,i+1 )V isubject to trV i = i ,I ≽ V i ≽ 0,(1701)i=1... ki=1... k
C.2. TRACE, SINGULAR AND EIGEN VALUES 659where Υ k+1,k+1 0. We speculate,k∑Υ ii λ(B) i = supi=1tr(ΥU T BU) = sup δ(Υ) T δ(U T BU) (1702)U∈ R N×kU∈ R N×kU T U=IU T U=Ik∑i=1Alizadeh shows: [11,4.2]k∑Υ ii λ(B) i = minimize iµ i + trZ iµ∈R k , Z i ∈S N i=1subject to µ i I + Z i − (Υ ii −Υ i+1,i+1 )B ≽ 0, i=1... k= maximizeV i ∈S Nwhere Υ k+1,k+1 0.Z i ≽ 0,()∑tr B k (Υ ii −Υ i+1,i+1 )V ii=1i=1... ksubject to trV i = i , i=1... kI ≽ V i ≽ 0, i=1... k (1703)The largest eigenvalue magnitude µ of A∈ S Nmax { |λ(A) i | } = minimize µi µ∈Rsubject to −µI ≼ A ≼ µI(1704)is minimized over convex set C by semidefinite program: (confer7.1.5)minimize ‖A‖ 2Asubject to A ∈ C≡minimize µµ , Asubject to −µI ≼ A ≼ µIA ∈ C(1705)id est,µ ⋆ maxi{ |λ(A ⋆ ) i | , i = 1... N } ∈ R + (1706)
- Page 607 and 608: A.3. PROPER STATEMENTS 607For A,B
- Page 609 and 610: A.3. PROPER STATEMENTS 609When B is
- Page 611 and 612: A.4. SCHUR COMPLEMENT 611A.4 Schur
- Page 613 and 614: A.4. SCHUR COMPLEMENT 613A.4.0.0.3
- Page 615 and 616: A.4. SCHUR COMPLEMENT 615From Corol
- Page 617 and 618: A.5. EIGENVALUE DECOMPOSITION 617wh
- Page 619 and 620: A.5. EIGENVALUE DECOMPOSITION 619A.
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- Page 625 and 626: A.7. ZEROS 625A.6.5SVD of symmetric
- Page 627 and 628: A.7. ZEROS 627(Transpose.)Likewise,
- Page 629 and 630: A.7. ZEROS 629For X,A∈ S M +[34,2
- Page 631 and 632: A.7. ZEROS 631A.7.5.0.1 Proposition
- Page 633 and 634: Appendix BSimple matricesMathematic
- Page 635 and 636: B.1. RANK-ONE MATRIX (DYAD) 635R(v)
- Page 637 and 638: B.1. RANK-ONE MATRIX (DYAD) 637B.1.
- Page 639 and 640: B.2. DOUBLET 639R([u v ])R(Π)= R([
- Page 641 and 642: B.3. ELEMENTARY MATRIX 641has N −
- Page 643 and 644: B.4. AUXILIARY V -MATRICES 643is an
- Page 645 and 646: B.4. AUXILIARY V -MATRICES 64514. [
- Page 647 and 648: B.5. ORTHOGONAL MATRIX 647Given X
- Page 649 and 650: B.5. ORTHOGONAL MATRIX 649Figure 15
- Page 651 and 652: B.5. ORTHOGONAL MATRIX 651which is
- Page 653 and 654: Appendix CSome analytical optimal r
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- Page 657: C.2. TRACE, SINGULAR AND EIGEN VALU
- Page 661 and 662: C.3. ORTHOGONAL PROCRUSTES PROBLEM
- Page 663 and 664: C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
- Page 665 and 666: C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
- Page 667 and 668: C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
- Page 669 and 670: Appendix DMatrix calculusFrom too m
- Page 671 and 672: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
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- Page 689 and 690: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
- Page 691 and 692: D.2. TABLES OF GRADIENTS AND DERIVA
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- Page 699 and 700: Appendix EProjectionFor any A∈ R
- Page 701 and 702: 701U T = U † for orthonormal (inc
- Page 703 and 704: E.1. IDEMPOTENT MATRICES 703where A
- Page 705 and 706: E.1. IDEMPOTENT MATRICES 705order,
- Page 707 and 708: E.1. IDEMPOTENT MATRICES 707are lin
C.2. TRACE, SINGULAR AND EIGEN VALUES 659where Υ k+1,k+1 0. We speculate,k∑Υ ii λ(B) i = supi=1tr(ΥU T BU) = sup δ(Υ) T δ(U T BU) (1702)U∈ R N×kU∈ R N×kU T U=IU T U=Ik∑i=1Alizadeh shows: [11,4.2]k∑Υ ii λ(B) i = minimize iµ i + trZ iµ∈R k , Z i ∈S N i=1subject to µ i I + Z i − (Υ ii −Υ i+1,i+1 )B ≽ 0, i=1... k= maximizeV i ∈S Nwhere Υ k+1,k+1 0.Z i ≽ 0,()∑tr B k (Υ ii −Υ i+1,i+1 )V ii=1i=1... ksubject to trV i = i , i=1... kI ≽ V i ≽ 0, i=1... k (1703)The largest eigenvalue magnitude µ of A∈ S Nmax { |λ(A) i | } = minimize µi µ∈Rsubject to −µI ≼ A ≼ µI(1704)is minimized over convex set C by semidefinite program: (confer7.1.5)minimize ‖A‖ 2Asubject to A ∈ C≡minimize µµ , Asubject to −µI ≼ A ≼ µIA ∈ C(1705)id est,µ ⋆ maxi{ |λ(A ⋆ ) i | , i = 1... N } ∈ R + (1706)
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