Linear programming quadratic convexity, and distance geometry

Conference · Sat Dec 31 04:00:00 EST 1994

OSTI ID:35810

Barvinok, A

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--
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
ALGORITHMS
EFFICIENCY
LINEAR PROGRAMMING
OPTIMIZATION