Convex analysis and variational problems
Convex analysis and variational problems
A relational model of data for large shared data banks
Communications of the ACM
Fundamentals of Database Systems
Fundamentals of Database Systems
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We examine the best approximation of componentwise positive vectors or positive continuous functions f by linear combinations f@?=@?"j@a"j@f"j of given vectors or functions @f"j with respect to functionals Q"p, 1@?p@?~, involving quotients max{f/f@?,f@?/f} rather than differences |f-f@?|. We verify the existence of a best approximating function under mild conditions on {@f"j}"j"="1^n. For discrete data, we compute a best approximating function with respect to Q"p, p=1,2,~ by second order cone programming. Special attention is paid to the Q"~ functional in both the discrete and the continuous setting. Based on the computation of the subdifferential of our convex functional Q"~ we give an equivalent characterization of the best approximation by using its extremal set. Then we apply this characterization to prove the uniqueness of the best Q"~ approximation for Chebyshev sets {@f"j}"j"="1^n.