v2009.01.01 - Convex Optimization

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7.0.3 Problem approach
Problems traditionally posed in terms of point position {x i ∈ R n , i=1... N }
such as
∑
minimize (‖x i − x j ‖ − h ij ) 2 (1213)
{x i }
or
minimize
{x i }
i , j ∈ I
∑
(‖x i − x j ‖ 2 − h ij ) 2 (1214)
i , j ∈ I
(where I is an abstract set of indices and h ij is given data) are everywhere
converted herein to the distance-square variable D or to Gram matrix 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 or
absolute position. This approach is taken because we prefer introduction
of rank constraints into convex problems rather than searching a googol of
local minima in nonconvex problems like (1213) or (1214) [89].
7.0.4 Three prevalent proximity problems
There are three statements of the closest-EDM problem prevalent in the
literature, the multiplicity due primarily to choice of projection on the
EDM versus positive semidefinite (PSD) cone and vacillation between the
distance-square variable d ij versus absolute distance √ d ij . In their most
fundamental form, the three prevalent proximity problems are (1215.1),
(1215.2), and (1215.3): [297]
(1)
(3)
minimize ‖−V (D − H)V ‖ 2 F
D
subject to rankV DV ≤ ρ
D ∈ EDM N
minimize ‖D − H‖ 2 F
D
subject to rankV DV ≤ ρ
D ∈ EDM N
minimize ‖ ◦√ D − H‖
◦√ 2 F
D
subject to rankV DV ≤ ρ (2)
◦√ √
D ∈ EDM
N
(1215)
minimize ‖−V ( ◦√ D − H)V ‖
◦√ 2 F
D
subject to rankV DV ≤ ρ
◦√ √
D ∈ EDM
N
(4)