v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
568 CHAPTER 7. PROXIMITY PROBLEMS7.2.1.1 Equivalent semidefinite program, Problem 2, convex caseConvex problem (1352) is numerically solvable for its global minimum usingan interior-point method [391] [289] [277] [382] [11] [145]. We translate (1352)to an equivalent semidefinite program (SDP) for a pedagogical reason madeclear in7.2.2.2 and because there exist readily available computer programsfor numerical solution [167] [384] [385] [359] [34] [383] [348] [336].Substituting a new matrix variable Y [y ij ]∈ R N×N+h ij√dij ← y ij (1354)Boyd proposes: problem (1352) is equivalent to the semidefinite program∑minimize d ij − 2y ij + h 2 ijD , Yi,j[ ]dij y ij(1355)subject to≽ 0 , i,j =1... Ny ij h 2 ijD ∈ EDM NTo see that, recall d ij ≥ 0 is implicit to D ∈ EDM N (5.8.1, (910)). Sowhen H ∈ R N×N+ is nonnegative as assumed,[ ]dij y √ij≽ 0 ⇔ h ij√d ij ≥ yij 2 (1356)y ijh 2 ijMinimization of the objective function implies maximization of y ij that isbounded above. √Hence nonnegativity of y ij is implicit to (1355) and, asdesired, y ij →h ij dij as optimization proceeds.If the given matrix H is now assumed symmetric and nonnegative,H = [h ij ] ∈ S N ∩ R N×N+ (1357)then Y = H ◦ ◦√ D must belong to K= S N h ∩ R N×N+ (1301). Because Y ∈ S N h(B.4.2 no.20), then‖ ◦√ D − H‖ 2 F = ∑ i,jd ij − 2y ij + h 2 ij = −N tr(V (D − 2Y )V ) + ‖H‖ 2 F (1358)
7.2. SECOND PREVALENT PROBLEM: 569So convex problem (1355) is equivalent to the semidefinite programminimize − tr(V (D − 2Y )V )D , Y[ ]dij y ijsubject to≽ 0 ,N ≥ j > i = 1... N −1y ij h 2 ij(1359)Y ∈ S N hD ∈ EDM Nwhere the constants h 2 ij and N have been dropped arbitrarily from theobjective.7.2.1.2 Gram-form semidefinite program, Problem 2, convex caseThere is great advantage to expressing problem statement (1359) inGram-form because Gram matrix G is a bidirectional bridge between pointlist X and distance matrix D ; e.g., Example 5.4.2.3.5, Example 6.7.0.0.1.This way, problem convexity can be maintained while simultaneouslyconstraining point list X , Gram matrix G , and distance matrix D at ourdiscretion.Convex problem (1359) may be equivalently written via linear bijective(5.6.1) EDM operator D(G) (903);minimizeG∈S N c , Y ∈ S N hsubject to− tr(V (D(G) − 2Y )V )[ ]〈Φij , G〉 y ij≽ 0 ,y ijh 2 ijN ≥ j > i = 1... N −1(1360)G ≽ 0where distance-square D = [d ij ] ∈ S N h (887) is related to Gram matrix entriesG = [g ij ] ∈ S N c ∩ S N + bywhered ij = g ii + g jj − 2g ij= 〈Φ ij , G〉(902)Φ ij = (e i − e j )(e i − e j ) T ∈ S N + (889)
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568 CHAPTER 7. PROXIMITY PROBLEMS7.2.1.1 Equivalent semidefinite program, Problem 2, convex case<strong>Convex</strong> problem (1352) is numerically solvable for its global minimum usingan interior-point method [391] [289] [277] [382] [11] [145]. We translate (1352)to an equivalent semidefinite program (SDP) for a pedagogical reason madeclear in7.2.2.2 and because there exist readily available computer programsfor numerical solution [167] [384] [385] [359] [34] [383] [348] [336].Substituting a new matrix variable Y [y ij ]∈ R N×N+h ij√dij ← y ij (1354)Boyd proposes: problem (1352) is equivalent to the semidefinite program∑minimize d ij − 2y ij + h 2 ijD , Yi,j[ ]dij y ij(1355)subject to≽ 0 , i,j =1... Ny ij h 2 ijD ∈ EDM NTo see that, recall d ij ≥ 0 is implicit to D ∈ EDM N (5.8.1, (910)). Sowhen H ∈ R N×N+ is nonnegative as assumed,[ ]dij y √ij≽ 0 ⇔ h ij√d ij ≥ yij 2 (1356)y ijh 2 ijMinimization of the objective function implies maximization of y ij that isbounded above. √Hence nonnegativity of y ij is implicit to (1355) and, asdesired, y ij →h ij dij as optimization proceeds.If the given matrix H is now assumed symmetric and nonnegative,H = [h ij ] ∈ S N ∩ R N×N+ (1357)then Y = H ◦ ◦√ D must belong to K= S N h ∩ R N×N+ (1301). Because Y ∈ S N h(B.4.2 no.20), then‖ ◦√ D − H‖ 2 F = ∑ i,jd ij − 2y ij + h 2 ij = −N tr(V (D − 2Y )V ) + ‖H‖ 2 F (1358)