v2007.11.26 - Convex Optimization

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680 APPENDIX F. MATLAB PROGRAMSbreakendendif ~violy(:,k) = v(:,1);elseif isempty(j)disp(’Sturm procedure taking default j’), j = 2; returnend % debugalpha = (-2*(v(:,1)’*A*v(:,j)) + sqrt((2*v(:,1)’*A*v(:,j)).^2 ...-4*(v(:,j)’*A*v(:,j))*(v(:,1)’*A*v(:,1))))/(2*(v(:,j)’*A*v(:,j)));y(:,k) = (v(:,1) + alpha*v(:,j))/sqrt(1+alpha^2);if chop(y(:,k)’*A*y(:,k),tol) ~= 0alpha = (-2*(v(:,1)’*A*v(:,j)) - sqrt((2*v(:,1)’*A*v(:,j)).^2 ...-4*(v(:,j)’*A*v(:,j))*(v(:,1)’*A*v(:,1))))/(2*(v(:,j)’*A*v(:,j)));y(:,k) = (v(:,1) + alpha*v(:,j))/sqrt(1+alpha^2);if chop(y(:,k)’*A*y(:,k),tol) ~= 0disp(’Zero problem in Sturm!’), returnend % debugendendY = Y - y(:,k)*y(:,k)’;rho = rho - 1;endz = zeros(size(y));e = zeros(N,N);for i=1:rz(:,i) = y(:,i)/norm(y(:,i));e(i,i) = norm(y(:,i))^2;endlam = diag(e);[junk id] = sort(real(lam));id = id(length(id):-1:1);z = [z(:,id(1:r)) null(z’)] % Sturme = diag(lam(id))[v,d] = signeig(X) % eigenvalue decompositionX-z*e*z’traceAX = trace(A*X)
F.6. CONVEX ITERATION DEMONSTRATION 681F.6 <strong>Convex</strong> Iteration demonstrationWe demonstrate implementation of a rank constraint in a semidefiniteBoolean feasibility problem from4.6.0.0.4. It requires CVX, [117] an intuitiveMatlab interface for interior-point method solvers.There are a finite number 2 N=50 ≈1E15 of binary vectors x . The feasibleset of semidefinite program (683) is the intersection of an elliptope withM =10 halfspaces in vectorized composite G . Size of the optimal rank-1solution set is proportional to the positive factor scaling vector b . Thesmaller that optimal Boolean solution set, the harder this problem is to solve;indeed, it can be made as small as one point. That scale factor and initialstate of random number generators, making matrix A and vector b , areselected to demonstrate Boolean solution to one instance in a few iterations(a few seconds), whereas sequential binary search takes one hour to test25.7 million vectors before finding one Boolean solution feasible to nonconvexproblem (680). (Other parameters can be selected to realize a reversal ofthese timings.)% Discrete optimization problem demo.% -Jon Dattorro, June 4, 2007% Find x\in{-1,1}^N such that Ax
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DATTORROCONVEXOPTIMIZATION&EUCLIDEA
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Convex Optimization&Euclidean Dista
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for Jennie Columba♦Antonio♦♦&
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PreludeThe constant demands of my d
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Convex Optimization&Euclidean Dista
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CONVEX OPTIMIZATION & EUCLIDEAN DIS
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List of Figures1 Overview 191 Orion
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LIST OF FIGURES 1559 Epigraph . . .
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LIST OF FIGURES 17120 Four fundamen
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Chapter 1OverviewConvex Optimizatio
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ˇx 4ˇx 3ˇx 2Figure 2: Applicatio
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23Figure 4: This coarsely discretiz
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ases (biorthogonal expansion). We e
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27Figure 7: These bees construct a
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its membership to the EDM cone. The
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31appendicesProvided so as to be mo
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Chapter 2Convex geometryConvexity h
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2.1. CONVEX SET 35Figure 11: A slab
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2.1. CONVEX SET 372.1.6 empty set v
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2.1. CONVEX SET 392.1.7.1 Line inte
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2.1. CONVEX SET 41(a)R 2(b)R 3(c)(d
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2.1. CONVEX SET 43This theorem in c
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2.2. VECTORIZED-MATRIX INNER PRODUC
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2.3. HULLS 53Figure 14: Convex hull
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2.3. HULLS 55Aaffine hull (drawn tr
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2.3. HULLS 57The union of relative
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2.4. HALFSPACE, HYPERPLANE 59of dim
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2.4. HALFSPACE, HYPERPLANE 61H +ay
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2.4. HALFSPACE, HYPERPLANE 63Inters
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2.4. HALFSPACE, HYPERPLANE 65Conver
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2.4. HALFSPACE, HYPERPLANE 67A 1A 2
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2.4. HALFSPACE, HYPERPLANE 69tradit
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2.4. HALFSPACE, HYPERPLANE 71There
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2.5. SUBSPACE REPRESENTATIONS 73nor
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2.5. SUBSPACE REPRESENTATIONS 752.5
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2.6. EXTREME, EXPOSED 77In other wo
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2.6. EXTREME, EXPOSED 792.6.1 Expos
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2.7. CONES 812.6.1.3.1 Definition.
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2.7. CONES 830Figure 26: Boundary o
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2.7. CONES 852.7.2 Convex coneWe ca
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2.7. CONES 87Thus the simplest and
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2.7. CONES 89nomenclature generaliz
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2.8. CONE BOUNDARY 91Proper cone {0
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2.8. CONE BOUNDARY 93the same extre
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2.8. CONE BOUNDARY 95For a proper c
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.10. CONIC INDEPENDENCE (C.I.) 121
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2.12. CONVEX POLYHEDRA 127It follow
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2.12. CONVEX POLYHEDRA 129Coefficie
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2.12. CONVEX POLYHEDRA 1312.12.3 Un
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2.12. CONVEX POLYHEDRA 133
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2.13. DUAL CONE & GENERALIZED INEQU
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Chapter 3Geometry of convex functio
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3.1. CONVEX FUNCTION 185f 1 (x)f 2
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3.1. CONVEX FUNCTION 1873.1.3 norm
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3.1. CONVEX FUNCTION 189n∑π(x) i
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3.1. CONVEX FUNCTION 191This means
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3.1. CONVEX FUNCTION 1933.1.5.3 pos
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3.1. CONVEX FUNCTION 1953.1.6.0.1 E
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3.1. CONVEX FUNCTION 1973.1.7 epigr
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3.1. CONVEX FUNCTION 201Although th
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3.1. CONVEX FUNCTION 203minimize tX
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3.1. CONVEX FUNCTION 2053.1.8 gradi
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3.1. CONVEX FUNCTION 209f(Y )[ ∇f
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3.1. CONVEX FUNCTION 211g is convex
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3.1. CONVEX FUNCTION 213αβα ≥
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3.1. CONVEX FUNCTION 2153.1.11 seco
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3.2. MATRIX-VALUED CONVEX FUNCTION
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3.2. MATRIX-VALUED CONVEX FUNCTION
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3.3. QUASICONVEX 221matrices in the
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3.3. QUASICONVEX 223on R M + . Alth
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Chapter 4Semidefinite programmingPr
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4.1. CONIC PROBLEM 227where K is a
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4.1. CONIC PROBLEM 229C0PΓ 1Γ 2S+
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4.1. CONIC PROBLEM 231faces of S 3
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4.1. CONIC PROBLEM 2334.1.1.3 Previ
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4.2. FRAMEWORK 235Equivalently, pri
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4.2. FRAMEWORK 237is positive semid
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4.2. FRAMEWORK 239Optimal value of
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4.2. FRAMEWORK 2414.2.3.1.1 Example
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4.2. FRAMEWORK 243where δ is the m
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4.2. FRAMEWORK 2454.2.3.1.2 Example
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4.3. RANK REDUCTION 2474.3 Rank red
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4.3. RANK REDUCTION 249A rank-reduc
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4.3. RANK REDUCTION 251(t ⋆ i)
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4.3. RANK REDUCTION 2534.3.3.0.1 Ex
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4.3. RANK REDUCTION 2554.3.3.0.2 Ex
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.5. CONSTRAINING CARDINALITY 2734.
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4.5. CONSTRAINING CARDINALITY 275Th
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4.5. CONSTRAINING CARDINALITY 277wh
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4.5. CONSTRAINING CARDINALITY 279f
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4.5. CONSTRAINING CARDINALITY 281al
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4.6. CARDINALITY AND RANK CONSTRAIN
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4.7. CONVEX ITERATION RANK-1 301whi
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4.7. CONVEX ITERATION RANK-1 303the
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Chapter 5Euclidean Distance MatrixT
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5.2. FIRST METRIC PROPERTIES 307cor
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5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
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5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
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5.4. EDM DEFINITION 313The collecti
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5.4. EDM DEFINITION 3155.4.2 Gram-f
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5.4. EDM DEFINITION 317D ∈ EDM N
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5.4. EDM DEFINITION 3195.4.2.2.1 Ex
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5.4. EDM DEFINITION 321ten affine e
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5.4. EDM DEFINITION 323spheres:Then
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5.4. EDM DEFINITION 325By eliminati
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5.4. EDM DEFINITION 327whereΦ ij =
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5.4. EDM DEFINITION 3295.4.2.2.5 De
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5.4. EDM DEFINITION 331105ˇx 4ˇx
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5.4. EDM DEFINITION 333corrected by
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5.4. EDM DEFINITION 335by translate
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5.4. EDM DEFINITION 337Crippen & Ha
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5.4. EDM DEFINITION 339where ([√t
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5.4. EDM DEFINITION 341because (A.3
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5.5. INVARIANCE 3435.5.1.0.1 Exampl
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5.6. INJECTIVITY OF D & UNIQUE RECO
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5.7. EMBEDDING IN AFFINE HULL 3515.
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5.7. EMBEDDING IN AFFINE HULL 353Fo
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5.7. EMBEDDING IN AFFINE HULL 3555.
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5.8. EUCLIDEAN METRIC VERSUS MATRIX
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5.9. BRIDGE: CONVEX POLYHEDRA TO ED
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5.10. EDM-ENTRY COMPOSITION 373(ii)
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5.11. EDM INDEFINITENESS 3755.11.1
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5.11. EDM INDEFINITENESS 377(confer
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5.11. EDM INDEFINITENESS 379we have
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5.11. EDM INDEFINITENESS 381For pre
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5.12. LIST RECONSTRUCTION 383where
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5.12. LIST RECONSTRUCTION 385(a)(c)
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5.13. RECONSTRUCTION EXAMPLES 387D
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5.13. RECONSTRUCTION EXAMPLES 389Th
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5.13. RECONSTRUCTION EXAMPLES 391wh
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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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Chapter 6Cone of distance matricesF
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6.1. DEFINING EDM CONE 4056.1 Defin
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6.2. POLYHEDRAL BOUNDS 407This cone
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6.3.√EDM CONE IS NOT CONVEX 409N
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6.4. A GEOMETRY OF COMPLETION 4116.
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6.4. A GEOMETRY OF COMPLETION 413(a
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6.4. A GEOMETRY OF COMPLETION 415Fi
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6.5. EDM DEFINITION IN 11 T 417and
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6.5. EDM DEFINITION IN 11 T 419then
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6.5. EDM DEFINITION IN 11 T 4216.5.
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6.5. EDM DEFINITION IN 11 T 423D =
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6.6. CORRESPONDENCE TO PSD CONE S N
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6.7. VECTORIZATION & PROJECTION INT
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6.7. VECTORIZATION & PROJECTION INT
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6.8. DUAL EDM CONE 435When the Fins
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6.8. DUAL EDM CONE 437Proof. First,
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6.8. DUAL EDM CONE 439EDM 2 = S 2 h
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6.8. DUAL EDM CONE 441whose veracit
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6.8. DUAL EDM CONE 4436.8.1.3.1 Exe
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6.8. DUAL EDM CONE 445has dual affi
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6.8. DUAL EDM CONE 4476.8.1.7 Schoe
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6.9. THEOREM OF THE ALTERNATIVE 449
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6.10. POSTSCRIPT 451When D is an ED
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Chapter 7Proximity problemsIn summa
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In contrast, order of projection on
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457HS N h0EDM NK = S N h ∩ R N×N
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4597.0.3 Problem approachProblems t
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7.1. FIRST PREVALENT PROBLEM: 461fi
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7.1. FIRST PREVALENT PROBLEM: 4637.
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7.1. FIRST PREVALENT PROBLEM: 465di
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7.1. FIRST PREVALENT PROBLEM: 4677.
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7.1. FIRST PREVALENT PROBLEM: 469wh
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7.1. FIRST PREVALENT PROBLEM: 471Th
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7.2. SECOND PREVALENT PROBLEM: 473O
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7.2. SECOND PREVALENT PROBLEM: 475S
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7.2. SECOND PREVALENT PROBLEM: 477r
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7.2. SECOND PREVALENT PROBLEM: 479c
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7.2. SECOND PREVALENT PROBLEM: 4817
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7.3. THIRD PREVALENT PROBLEM: 483fo
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7.3. THIRD PREVALENT PROBLEM: 485a
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7.3. THIRD PREVALENT PROBLEM: 4877.
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7.3. THIRD PREVALENT PROBLEM: 4897.
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7.3. THIRD PREVALENT PROBLEM: 491Ou
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7.4. CONCLUSION 493The rank constra
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Appendix ALinear algebraA.1 Main-di
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A.1. MAIN-DIAGONAL δ OPERATOR, λ
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A.2. SEMIDEFINITENESS: DOMAIN OF TE
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A.2. SEMIDEFINITENESS: DOMAIN OF TE
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A.3. PROPER STATEMENTS 503A.3.0.0.1
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A.3. PROPER STATEMENTS 505By simila
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A.3. PROPER STATEMENTS 507Because R
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For A,B ∈ R n×n x T Ax ≥ x T B
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A.3. PROPER STATEMENTS 511A.3.1.0.2
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A.3. PROPER STATEMENTS 513We can de
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A.4. SCHUR COMPLEMENT 515Origin of
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A.4. SCHUR COMPLEMENT 517Schur-form
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A.5. EIGEN DECOMPOSITION 519A.5.0.1
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A.6. SINGULAR VALUE DECOMPOSITION,
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A.7. ZEROS 527For diagonalizable ma
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A.7. ZEROS 529A.7.4For X,A∈ S M +
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A.7. ZEROS 531A.7.5.0.1 Proposition
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Appendix BSimple matricesMathematic
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B.1. RANK-ONE MATRIX (DYAD) 535R(v)
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B.1. RANK-ONE MATRIX (DYAD) 537rang
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B.2. DOUBLET 539R([u v ])R(Π)= R([
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B.3. ELEMENTARY MATRIX 541If λ ≠
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B.4. AUXILIARY V -MATRICES 543the n
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B.4. AUXILIARY V -MATRICES 54518. V
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B.5. ORTHOGONAL MATRIX 547B.5 Ortho
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B.5. ORTHOGONAL MATRIX 549B.5.3.0.1
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Appendix CSome analytical optimal r
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C.2. DIAGONAL, TRACE, SINGULAR AND
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C.3. ORTHOGONAL PROCRUSTES PROBLEM
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C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
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C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
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Appendix DMatrix calculusFrom too m
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D.1. DIRECTIONAL DERIVATIVE, TAYLOR
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D.2. TABLES OF GRADIENTS AND DERIVA
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Appendix EProjectionFor any A∈ R
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597Equivalent to the corresponding
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E.1. IDEMPOTENT MATRICES 599where R
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E.1. IDEMPOTENT MATRICES 601TxT ⊥
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E.2. I − P , PROJECTION ON ALGEBR
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E.3. SYMMETRIC IDEMPOTENT MATRICES
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E.5. PROJECTION EXAMPLES 611E.5.0.0
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E.5. PROJECTION EXAMPLES 613of rela
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E.5. PROJECTION EXAMPLES 615E.5.0.0
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E.6. VECTORIZATION INTERPRETATION,
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E.7. ON VECTORIZED MATRICES OF HIGH
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E.7. ON VECTORIZED MATRICES OF HIGH
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Page 633 and 634: E.9. PROJECTION ON CONVEX SET 633Pr
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Page 661 and 662: Appendix FMatlab programsMade by Th
Page 663 and 664: F.1. ISEDM() 663% is nonnegativeif
Page 665 and 666: F.1. ISEDM() 665F.1.1Subroutines fo
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Page 671 and 672: F.3. MAP OF THE USA 671% plot origi
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Page 679: F.5. STURM’S PROCEDURE 679F.5 Stu
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Page 689 and 690: Appendix GNotation and a few defini
Page 691 and 692: 691l.i.w.r.tlinearly independentwit
Page 693 and 694: 693t → 0 +t goes to 0 from above;
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Page 701 and 702: 701maximizexargsup Xarg supf(x)subj
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Page 705 and 706: Bibliography[1] Suliman Al-Homidan
Page 707 and 708: BIBLIOGRAPHY 707[16] Keith Ball. An
Page 709 and 710: BIBLIOGRAPHY 709[38] A. W. Bojanczy
Page 711 and 712: BIBLIOGRAPHY 711[57] Steven Chu. Au
Page 713 and 714: BIBLIOGRAPHY 713[76] Frank R. Deuts
Page 715 and 716: BIBLIOGRAPHY 715(AACC), June 2004.h
Page 717 and 718: BIBLIOGRAPHY 717[114] John Clifford
Page 719 and 720: BIBLIOGRAPHY 719[135] Uwe Helmke an
Page 721 and 722: BIBLIOGRAPHY 721[157] Joakim Jaldé
Page 723 and 724: BIBLIOGRAPHY 723[178] Jung Rye Lee.
Page 725 and 726: BIBLIOGRAPHY 725[200] T. S. Motzkin
Page 727 and 728: BIBLIOGRAPHY 727[220] Teemu Pennane
Page 729 and 730: BIBLIOGRAPHY 729[240] Joshua A. Sin
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BIBLIOGRAPHY 731[264] Warren S. Tor
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BIBLIOGRAPHY 733[285] Bernard Widro
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Index0-norm, 241, 273, 275, 2941-no
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INDEX 737clipping, 190, 454, 638, 6
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INDEX 739hull, 53, 56, 306of outer
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INDEX 741unique, 519elbow, 457, 458
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INDEX 743polarity, 62handoff, 335ov
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INDEX 745manifold, 23, 24, 105, 410
Page 747 and 748:
INDEX 747objectivebilinear, 256, 27
Page 749 and 750:
INDEX 749Hadamard, 46, 472, 571, 69
Page 751 and 752:
INDEX 751trace, 477, 478heuristic,
Page 753 and 754:
INDEX 753symmetric hollow, 51tangen
Page 755 and 756:
INDEX 755matrix, 65slack, 227vec, 4
Page 757 and 758:
757
Page 760:
Convex Optimization & Euclidean Dis
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v2007.11.26 - Convex Optimization

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