Linear programming quadratic convexity, and distance geometry
We apply Linear Programming in the space of quadratic forms (known also as Semidefinite Programming) to investigate when the image of a quadratic map q: R{sup n} {yields} R{sup k} is convex. With a given quadratic map q: R{sup n} {yields} R{sup k} and a number d one can naturally associate a quadratic map qd: R{sup nd} {yields} R{sup k} by tensoring the matrices of the map q onto the d {times} d identity matrix. We show that the image of qd is convex provided d {>=} (({radical}8k + 1-1)/2) (this bound is sharp in the worst case). This result allows us to obtain extensions of the Toeplitz-Hausdorff convexity theorem and to reveal {open_quotes}hidden convexity{close_quotes} of many difficult problems in Distance Geometry. For example, the problem of finding whether it is possible to realize a given weighted graph with k edges as a graph of distances between some points in R{sub d} is a problem of Semidefinite Programming provided d {>=} [({radical}8k + 1 - 1)/2].
- OSTI ID:
- 35810
- Report Number(s):
- CONF-9408161-; TRN: 94:009753-0068
- Resource Relation:
- Conference: 15. international symposium on mathematical programming, Ann Arbor, MI (United States), 15-19 Aug 1994; Other Information: PBD: 1994; Related Information: Is Part Of Mathematical programming: State of the art 1994; Birge, J.R.; Murty, K.G. [eds.]; PB: 312 p.
- Country of Publication:
- United States
- Language:
- English
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