v2007.11.26 - Convex Optimization

4597.0.3 Problem approachProblems traditionally posed in terms of point position {x i , i=1... N } ,such as∑minimize (‖x i − x j ‖ − h ij ) 2 (1140){x i }orminimize{x i }i , j ∈ I∑(‖x i − x j ‖ 2 − h 2 ij) 2 (1141)i , j ∈ I(where I is an abstract set of indices and h ij is given data) are everywhereconverted (in this book) to the distance-square variable D or to Grammatrix G ; the Gram matrix acting as bridge between position and distance.That conversion is performed regardless of whether known data is complete.Then the techniques of chapter 5 or chapter 6 are applied to find relative orabsolute position. This approach is taken because we prefer introduction ofrank constraints into convex problems rather than searching an infinitude oflocal minima in (1140) or (1141) [70].7.0.4 Three prevalent proximity problemsThere are three statements of the closest-EDM problem prevalent in theliterature, the multiplicity due primarily to choice of projection on theEDM versus positive semidefinite (PSD) cone and vacillation between thedistance-square variable d ij versus absolute distance √ d ij . In their mostfundamental form, the three prevalent proximity problems are (1142.1),(1142.2), and (1142.3): [260](1)(3)minimize ‖−V (D − H)V ‖ 2 FDsubject to rankV DV ≤ ρD ∈ EDM Nminimize ‖D − H‖ 2 FDsubject to rankV DV ≤ ρD ∈ EDM Nminimize ‖ ◦√ D − H‖◦√ 2 FDsubject to rankV DV ≤ ρ (2)◦√ √D ∈ EDMN(1142)minimize ‖−V ( ◦√ D − H)V ‖◦√ 2 FDsubject to rankV DV ≤ ρ◦√ √D ∈ EDMN(4)